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Gravitation  versus  Relativity 

A  Non-Technical  Explanation  of  the  Fundamental  Principles  of 

Gravitational  Astronomy  and  a  Critical  Examination  of 

the  Astronomical  Evidence  Cited  as  Proof  of 

the  Generalized  Theory  of  Relativity 


By 
Charles  Lane  Poor 

Professor  of  Celestial  Mechanics  in  Columbia  University;  Author  of 
'The  Solar  System,"  "Nautical  Science,"  "Simpli- 
fied Navigation,"  etc. 


With  a  Preliminary  Essay  by 
Thomas  Chrowder  Chamberlin 

Emeritus  Professor  of  Geology  in  the  University  of  Chicago;  Senior 

Editor  of  the  Journal  of  Geology;  Autlwr  of  "  The  Origin 

of  the  Earth,"  and  other  geological  works 


Illustrated 


G.  P.  Putnam's  Sons 

New  York  London 

3be  "Knickerbocker  jpress 
1922 


Copyright,  1922 

by 
Charles  Lane  Poor 

Made  in  the  United  States  of  America 


ENGINEERING  LIBRARY 


AUTHOR'S  PREFACE 

THIS  work  is  intended  for  the  general  reader  as  well 
as  for  the  scientist,  working  in  lines  other  than  astron- 
omy. It  is  an  attempt  to  explain  in  non-technical  lan- 
guage and  without  the  use  of  complicated  mathematical 
formulas  the  fundamental  facts  and  principles  of  gravi- 
tational astronomy,  and  to  submit  the  so-called  astro- 
nomical proofs  of  the  Relativity  Theory  to  a  critical 
examination  and  discussion.  The  validity  of  these 
proofs  cannot  be  passed  upon  by  one  who  is  totally 
unfamiliar  with  the  facts  and  methods  of  astronomical 
research.  It  is  not,  however,  necessary  for  one  to 
know  all  the  complicated  details  of  planetary  motion, 
nor  to  be  familiar  with  all  the  methods  of  determining 
the  size,  shape,  and  motions  of  Mercury,  but  it  is  essen- 
tial for  one,  who  wishes  fairly  to  judge  the  evidence, 
to  know  the  fundamental  methods  and  approximations 
used  in  determining  the  motion  of  a  planet  about  the 
sun.  To  un familiarity  with  these  methods  may  be 
traced  a  widespread  misconception  as  to  the  intrinsic 
value  of  the  evidence. 

Further,  in  the  way  in  which  this  evidence  has  been 
presented  and  accepted,  there  has  been  apparently  a 
complete  reversal  of  ordinary  scientific  methods.  As  a 
new  theory,  as  an  hypothesis  seeking  acceptance,  it 

iii 

888761 


iv  Author's  Preface 

would  seem  that  the  burden  of  proof  should  rest  upon 
Relativity;  that  its  advocates  should  conclusively  prove 
the  necessity  and  the  sufficiency  of  their  hypothesis. 
Such,  at  least,  has  been  the  accepted  scientific  method 
in  the  past.  In  1665  Isaac  Newton  developed  his  law 
of  gravitation  and  put  it  to  test  in  the  motion  of  the 
moon.  He  found  a  minute  discrepancy  between  his 
theory  and  the  actual  motion  of  the  moon,  a  trifle  less 
than  one  one-hundred-and-thirtieth  (i/i3Oth)  of  an 
inch  in  a  second  of  time;  a  difference  of  about  i$% 
of  the  observed  motion.  This  small  discordance  caused 
Newton  to  consider  his  theory  as  not  proved,  and 
he  laid  aside  his  work.  Nearly  twenty  years  later 
new  measurements  of  the  earth  were  made,  and,  urged 
on  by  Halley,  Newton  corrected  his  older  calculations 
and  showed  that  his  law  of  gravitation  was  substan- 
tially correct.  Then,  and  then  only,  did  he  announce 
his  theories.  It  was  some  twenty  years  after  Charles 
Darwin  first  conceived  his  theory  of  evolution  before 
he  made  it  public  in  his  classic  work. 

The  theory  of  relativity,  on  the  contrary,  was  an- 
nounced without  any  confirmation.  Tests  were  pro- 
posed, selected  by  its  author,  and  these  tests  failed  of 
confirmation  by  20%,  by  50%  even,  and  yet  such 
results  are  called  thoroughly  satisfactory.  The  merest 
indication  of  a  result,  favorable  to  relativity,  becomes 
conclusive  proof;  and  observations  and  experiments, 
which  can  be  explained  by  the  new  hypothesis  almost 


Author's  Preface  v 

as  well  as  by  the  older  methods,  become  crucial  tests  in 
favor  of  relativity.  And  the  theory  has  been  accepted, 
and  is  accepted  by  mathematicians,  by  physicists,  by 
many  of  the  most  prominent  astronomers  of  the  world, 
and  the  burden  of  proof  has  been  shifted,  until 
it  seems  that  relativity  is  an  established  scientific  fact, 
unless  it  can  be  completely  disproved. 

For  some  years  now  the  entire  world  has  been  in  a 
state  of  unrest;  mental  as  well  as  physical.  The  physi- 
cal aspects  of  this  unrest,  the  strikes,  the  socialistic  up- 
risings, the  war,  are  vivid  memories;  the  deep  mental 
disturbances  are  evidenced  by  the  widespread  interest  in 
social  problems,  by  the  futuristic  movements  in  art,  by 
the  light  and  easy  way  in  which  many  cast  aside  the 
well  tested  theories  of  finance  and  government  in  favor 
of  radical  and  untried  experiments.  Can  it  be  that  the 
same  spirit  of  unrest  has  invaded  science?  The  Rela- 
tivity Theory,  as  announced  by  Einstein,  shatters  our 
fundamental  ideas  in  regard  to  space  and  time, 
destroys  the  basis  upon  which  has  been  built  the  entire 
edifice  of  modern  science,  and  substitutes  a  nebulous 
conception  of  varying  standards  and  shifting  unreal- 
ities. And  this  radical,  this  destroying  theory  has 
been  accepted  as  lightly  and  as  easily  as  one  accepts 
a  correction  to  the  estimated  height  of  a  mountain  in 
Asia,  or  to  the  source  of  a  river  in  equatorial  Africa. 

The  bases  of  our  fundamental  concepts  of  time  and 
space,  and  the  psychological  phases  of  the  Relativity 


vi  Author's  Preface 

Theory  are  but  lightly  touched  on  in  this  work,  and 
then  only  when  necessary  to  the  clear  presentation  of 
the  main  subject.  But  these  aspects  of  the  theory  have 
been  most  ably  presented  in  the  PRELIMINARY 
ESSAY  by  Professor  Chamberlin,  where  the  gradual 
evolution  of  our  inherited  fundamental  concepts  is 
traced  through  the  eons  of  geological  time,  and  the 
solid  bases  of  these  concepts  contrasted  with  the 
shadowy  structure  of  Minkowski  and  Einstein. 

It  is  the  main  purpose  of  this  book  to  present  to  the 
jury  of  the  thinking  world  the  concrete  astronomical 
evidence  cited  by  Einstein  and  the  relativitists  as  proof 
of  the  Generalized  Theory  of  Relativity,  and  to  sub- 
ject that  evidence  to  a  critical  examination.  Many  a 
well-built-up  case  has  completely  collapsed  under  a 
searching  examination  of  the  evidence  and  a  cross- 
examination  of  its  chief  witnesses. 

The  thanks  of  the  author  are  due  to  Dr.  George 
E.  Hale,  Director  of  the  Mount  Wilson  Observatory, 
to  Professor  Dayton  C.  Miller,  of  the  Case  School 
of  Applied  Science,  and  to  others  for  furnishing 
original  photographs  and  drawings  to  illustrate  the 
text;  to  Professors  Bergen  Davis  and  Harold  W. 
Webb  of  Columbia  University;  to  Mr.  William  E. 
Spandow,  who  read  and  corrected  the  text  and  com- 
piled the  index. 

C.  L.  P. 

DERING  HARBOR,  N.  Y., 
August,  1922. 


A  PRELIMINARY  ESSAY  UPON  THE  FUNDA- 
MENTAL CONCEPTS  OF  TIME  AND  SPACE 

AT  no  stage  of  history  has  equipoise  in  thinking 
been  more  vital  to  the  welfare  of  mankind  than  at  the 
present  time.  To  a  degree  perhaps  never  equalled  be- 
fore, the  social,  religious,  political,  economic,  and  most 
other  maxims  that  have  served  as  guides  in  the  past 
are  receiving  scant  deference  and  are  often  suffering 
open  question  or  active  hostility.  With  this  there  has 
come  a  general  loosening  of  restraints  and  an  unpre- 
cedented venturesomeness  into  untried  lines  of  thought, 
feeling  and  action.  While  all  this  is  to  be  viewed  with 
steady  vision  and  philosophic  calm — because  beyond 
question  there  is  in  it  much  that  is  good  as  well  as 
much  that  is  bad — its  import  is  so  grave  as  to  call  for 
serious  consideration.  Millions  have  already  perished 
unnecessarily;  many  more  millions  have  suffered  need- 
lessly ;  still  other  millions  are  on  the  brink  of  calamity. 
On  the  other  hand,  perhaps  even  greater  multitudes 
are  rising — in  spite  of  the  turmoil — to  higher  planes 
of  intellectual  and  ethical  action.  It  is  no  part  of  the 
function  of  this  little  essay  to  balance  the  ledger 
of  good  and  ill;  that  would  be  a  formidable  under- 
taking. It  is  merely  its  privilege  to  try  to  veer  the 
trend  of  thought  a  little  in  the  direction  of  restraint 
and  circumspection. 

vii 


Vlll 


A  Preliminary  Essay 


The  present  loosening  of  ties  and  venturesome  drift 
is  not  confined  to  the  strenuous  affairs  of  life  just  now 
distraught  by  extraordinary  conditions;  it  reaches 
down  into  the  fundamentals  of  thought  and  touches  the 
intellectual  instincts  inherited  from  the  great  past.  In 
particular,  the  basal  concepts  of  space  and  time,  the 
very  framework  of  thought,  are  being  called  in 
question.  Space  has  commonly  been  pictured  as  an 
unbounded  receptacle  for  all  that  is  and  all  that  takes 
place,  and  time,  as  the  tally  sheet  of  the  onsweep  of 
an  active  world.  Free  room  for  the  great  deployments 
of  the  cosmos  have  been  thought  to  be  offered  by 
unlimited  space  and  ample  duration  for  their  evolu- 
tions in  unrestricted  time.  The  world's  chief  interest 
has  always  lain  in  the  entities  acting  in  space  and 
time  rather  than  in  space  and  time  themselves,  but 
these  have  been  felt  to  be  none  the  less  vital  as 
necessary  conditions  for  the  work  of  the  positive 
agencies.  We  are  now  asked  to  cast  these  great 
basal  concepts  aside  and  view  space  and  time 
as  dependencies  tied  up  with  one  another  and  with 
the  very  entities  and  activities  heretofore  pictured  as 
playing  their  parts  within  them.  This  seems  to  trans- 
form the  whole  into  an  intertanglement  of  variables 
and  relativities  devoid  of  a  stable  groundwork  of  in- 
herent realities.  Thus  the  very  roots  of  the  thinking 
of  the  ages  are  being  digged  about  with  little  show 
of  care  whether  it  will  promote  growth  or  lead  to 
withering.  Specifically,  it  is  affirmed  that  space  and 
time,  far  from  being  boundless  and  independent,  are 
so  bound  to  each  other  that  they  have  no  independent 
existence,  that  all  motion  is  merely  relative,  that  there 


A  Preliminary  Essay  ix 

is  no  absolute  motion,  that  our  measuring-rods  are 
variable  according  as  they  move  in  one  dimension  or 
another,  and  the  rate  of  our  time-keepers  faster  or 
slower  according  as  they  move  at  one  velocity  or  an- 
other. It  is  thus  insisted  that  the  very  elements  of 
thought  need  reconstruction.  It  is  urged  that  the 
geometry  of  Euclid,  the  dynamics  of  Galileo,  and  the 
celestial  mechanics  of  Newton  are  basally  defective. 

All  these  revolutionary  claims  are  put  forth,  of 
course,  in  the  name  of  progress.  No  doubt  progress 
will  follow  this  unsettlement  of  ideas  when  a  new 
settlement  shall  be  reached,  whatever  it  shall  be,  but 
loss  and  damage  are  also  liable  to  attend  such  a  dis- 
turbance and  reorganization.  The  vital  question  of 
the  hour  is  how  to  preserve  inherited  values  and  add 
to  them  the  greatest  possible  measure  of  new  values, 
with  the  least  possible  adventure  into  what  is  futile 
or  harmful.  The  world  has  reason  to  be  proud  of 
its  recent  advances  in  knowledge  but  it  cannot  blink 
the  fact  that  there  has  also  been  an  increase  in  ways 
of  doing  wrong  and  thinking  foolishly.  Quite  as- 
suredly the  world  knows  much  more  now  than  in  the 
days  of  Aristotle,  but  it  is  equally  certain  that  it  knows 
many  more  ways  of  making  mistakes.  Still,  in  the 
face  of  all  the  menacing  entanglements  of  good  and 
ill,  it  seems  clearly  the  part  of  wisdom  to  push  ahead, 
but  it  seems  quite  as  clearly  the  part  of  folly  to  plunge 
into  the  untried  without  forethought  and  restraint. 
The  law  of  wisdom  is  to  test  first,  and  put  into  the 
foundations  afterward. 

The  great  thinking  public  is  not  so  much  concerned 
with  the  ultra-refined  accuracy  or  inaccuracy  of  parti c- 


x  A  Preliminary  Essay 

ular  systems  of  geometry,  mechanics  or  dynamics  as 
held  by  the  great  men  of  the  past,  as  with  the  soundness 
or  unsoundness  of  the  fundamental  modes  of  thought 
inherited  from  the  past.  Thinking  men  are  profoundly 
interested  in  the  question  whether  critical  inspection 
today  shows  that  the  foundation  stones  of  the  intel- 
lectual structures  thus  far  built  are  solid  in  substance 
and  essence — though  quite  certainly  affected  by  in- 
felicities in  selection,  cutting,  trimming  and  fitting — 
or  whether  such  inspection  shows  that  flaws  and 
fissures  seriously  weaken  the  foundation  stones  and 
require  their  replacement  before  any  higher  super- 
structures are  built  upon  them. 

The  first  and  most  basal  question  therefore  con- 
cerns the  way  these  modes  of  thought  have  come  into 
being  and  what  sanctions  they  bring  because  of  this 
origin.  Under  the  evolutionary  view,  our  basal  modes 
of  thought  have  grown  up  by  test  and  trial  out  of 
the  crucial  experiences  of  the  long  past;  not  from 
human  experience  alone  but  from  the  tests  and  trials 
of  the  long  line  of  living  beings  that  formed  the  human 
ancestry.  Out  of  these  experiences  have  come  the 
instinctive  reactions  that  guard  our  physical  welfare 
and  the  mental  reactions  once  called  self-evident  or 
axiomatic  truths.  The  question  then  follows :  If 
the  mental  processes  of  the  thinking  and  feeling 
world  are  thus  the  products  of  the  tests  and  trials 
of  the  ages,  must  they  not  be  in  line  with  the  realities 
of  nature?  Is  it  possible  that  a  system  of  basally 
false  thought,  feeling  and  action  has  escaped  disaster 
and  has  even  guided  evolution  upward  through  hun- 
dreds of  millions  of  years?  The  question  is  not 


A  Preliminary  Essay  xi 

whether  modes  of  thinking  free  from  shortcomings, 
mistakes,  illusions  and  even  serious  errors  have  been 
evolved,  but  whether  basal  soundness  lies  beneath  them 
rather  than  fundamental  falsity. 

Now  perceptions  of  space  and  time — as  well  as  action 
based  on  such  perceptions — have  been  matters  of  life 
and  death  to  each  of  many  generations  of  perceptive 
beings  since  the  trilobites  of  the  Cambrian  seas  gave 
chase  or  were  chased  and  brought  into  service  such 
perceptions  of  space  and  time  as  they  then  possessed. 
At  least  as  early  as  this,  eyes  and  other  sense  organs 
concerned  in  the  perception  of  space  and  time  had 
been  developed  to  help  in  pursuit  or  in  escape.  These 
have  been  greatly  sharpened  in  the  course  of  subsequent 
ages.  When  a  hawk  plunges  toward  a  coveted  bird 
and  the  bird  scuds  away  with  his  utmost  speed,  veer- 
ing his  course  this  way  and  that  to  escape,  there  come 
into  play  keen  perceptions  and  quick  uses  of  space  and 
time  as  well  as  the  relativities  of  pursuer  and  pursued. 
In  such  sharp  contests  it  is  quite  essential  to  see 
whether  the  field  is  occupied  with  filled  or  unfilled 
space,  for  the  latter  alone  is  available  for  action.  Out 
of  the  multitude  of  such  critical  actions  grew  the 
strong  working  sense  of  space  so  prevalent  in  the 
living  world. 

The  sense  of  time  appears  to  have  grown  up  in 
close  relation  to  that  of  space  in  so  far  as  action  was 
a  part  of  the  living  experience,  but  when  action  ceased, 
time  seemed  to  be  independent  of  space,  for  time  ap- 
peared to  roll  on  while  space  remained  unchanged. 
Thus  a  sense  of  the  essential  independence  of  time 
and  space  naturally  grew  out  of  working  experience. 


xii  A  Preliminary  Essay 

These  mental  reactions  first  appeared  in  the  ancestry 
of  man,  but  in  the  course  of  the  ages  they  were  con- 
tinued onward  and  upward  into  the  reactions  of  man 
himself,  and  became  firmly  fixed  in  his  mental  con- 
stitution as  hereditary,  organic  or  instinctive  modes  of 
mental  action. 

Along  with  these  instinctive  perceptions  of  the  space 
that  surrounded  the  evolving  beings  were  the  not  less 
fundamental  perceptions  of  extension  in  their  own 
bodies.  The  hand,  the  foot  and  the  mouth  were  easily 
perceived  to  be  separate  in  space,  and  were  also  found 
to  be  separable  in  various  degrees  at  will,  while  the 
time  factor  was  small  or  nil  and  was  subject  to  a 
different  set  of  variations  at  will.  Thus  the  will  habit- 
ually adjusts  time  to  space  and  space  to  time  on  the 
instinctive  perception  of  their  independence.  All  the 
senses  played  their  parts  in  these  practical  combinations 
of  space  and  time.  Each  sense  tested  the  perceptions 
of  the  others.  Their  combined  testimonies  give  un- 
faltering conviction  of  their  essential  soundness. 

During  the  ascent  of  man,  these  deep-seated,  organic, 
instinctive  perceptions  furnished  material  that  was 
built  into  the  natural  sciences.  Space  relations  on  the 
face  of  the  earth  gave  subject-matter  for  geography, 
topography,  geology,  geodesy,  and  the  other  earth 
sciences.  Space  measurements  in  the  laboratories  gave 
precise  material  for  the  mechanical,  physical,  and 
chemical  sciences.  The  relations  of  space  and  time  in 
the  lower  heavens  entered  into  meteorology,  while  the 
vast  relations  of  time  and  space  in  the  outer  heavens  gave 
material  for  astronomy  and  astrophysics.  All  these 
sciences  thus  rest  intimately  on  ideas  of  space  as  the 


A  Preliminary  Essay  xiii 

receptacle  and  natural  frame  of  reference  of  all  cosmic 
things,  and  on  ideas  of  time  as  the  tally  sheet  of 
successive  events.  The  vital  point  here  urged  is  that 
our  ideas  of  space  and  time  have  deep  organic  rootage. 
They  were  not  devised  by  Euclid,  Galileo,  or  Newton. 
They  were  inherited  by  these  master  thinkers,  whose 
contribution  to  us  has  been  a  clear  and  serviceable 
formulation  of  these  ideas.  It  is  not  the  personal 
views  of  these  great  men  that  are  called  in  question  so 
much  as  the  instinctive  organic  reactions  of  the  human 
race. 

But  strong  as  is  this  argument  that  a  system  of 
mental  reaction  evolved  from  the  tests  and  trials  of 
many  millions  of  years  is  fundamentally  sound,  it  does 
not  reach,  or  even  closely  approach,  inerrancy  in  details 
or  even  in  vital  matters;  much  less  does  it  imply  that 
the  highest  attainment  has  been  reached.  The  very 
scheme  of  evolution  implies  indefinite  struggle  for 
closer  adaptation  of  the  active  agents  to  the  conditions 
under  which  they  act.  This  leaves  an  open  invitation 
for  inquiry  in  all  directions  in  the  hope  of  reaching 
something  that  fits  more  closely  the  essential  working 
realities.  The  argument  does,  however,  carry  the 
admonition  that  advocates  of  new  adventures  which 
strike  at  the  roots  of  things  should  expect  to  find  tre- 
mendous odds  arrayed  against  them  and  should  there- 
fore put  their  views  to  long,  severe  and  patient  tests 
before  assuming  that  they  are  true  and  before  asking 
acceptance  or  beginning  propagandism.  If  they  under- 
take this  they  should  give  elaborate  and  explicit  recog- 
nition of  the  old,  and  equally  explicit  expositions  of  the 
new. 


XIV 


A  Preliminary  Essay 


True  scientists  have  felt  impelled  all  along  to  strive 
earnestly  for  the  utmost  precision  wherever  precision 
is  important.  As  a  result  of  persistent  and  scrupulous 
care,  the  existing  sciences  have  been  rewarded  with 
great  triumphs  of  precision.  These  serve  as  verifi- 
cations of  their  fundamental  trustworthiness.  To  the 
public,  the  prediction  of  eclipses  stands  forth  as  a 
most  signal  triumph  and  a  verification  of  the  New- 
tonian mechanics. 

It  is  eminently  proper,  however,  to  challenge  even 
these  triumphs,  but  the  challenge  of  the  relativists  has 
not  lain  against  the  trustworthiness  of  the  Newtonian 
prediction  of  eclipses,  nor  against  familiar  triumphs 
of  precision  in  other  standard  lines.  It  has  been  con- 
fined thus  far  to  a  few  little  known  discrepancies  of 
a  very  minute  sort.  And  so  Dr.  Poor  has  found  it 
obligatory  to  set  forth  with  great  care  and  precision 
the  fundamental  facts  and  principles  necessary  to  a 
full  and  judicial  opinion  on  the  merits  or  lack  of  merit 
of  the  claims  of  the  new  views.  It  is  an  essential  part 
of  the  purpose  of  this  book  to  meet  the  challenge  of  the 
Einstein  Theory  of  Relativity  on  its  own  selected 
grounds  in  so  far  as  these  are  astronomical. 

To  fully  appreciate  the  bearings  of  the  Einstein  view 
of  relativity,  it  is  well  to  recall  the  growth  of  the  idea 
of  relativity  in  the  usual  sense  of  the  term,  for  rela- 
tivity is  nothing  new.  When  a  Cambrian  trilobite 
chased  some  other  -ite  or  was  chased  by  it,  there  came 
into  play  relativities  of  space,  time,  strength,  speed  and 
skill.  The  great  organic  struggle  of  the  ages  from 
beginning  to  end  was  an  intricate  tangle  of  relativities. 
Moreover,  these  were  constantly  changing.  As  soon 


A  Preliminary  Essay  Xv 

as  thinking  reached  a  discriminating  stage,  it  appeared 
that  where  there  were  relations  there  must  be  things 
to  be  related.  And  so,  while  relativities  were  taken  into 
account,  they  were  regarded  as  dependent  on  inherent 
qualities  that  were  not  merely  relative.  The  effort  was 
to  keep  the  balance  between  that  which  was  dependent 
on  relationships  and  that  which  would  remain  if  the 
relationships  were  eliminated.  Even  causal  relation- 
ships are  extremely  numerous  and  highly  changeable. 
For  example,  following  the  Newtonian  doctrine  that 
gravitation  is  universal,  the  motions  of  the  earth  are 
effected  by  the  relative  positions  of  all  the  other  bodies 
in  the  heavens ;  its  relativities  of  motion  thus  number 
hundreds  of  millions,  and  one  set  follows  another  every 
minute.  It  is  absolutely  impossible  to  deal  with  all  or 
ascertain  what  is  their  sum  total.  Hundreds  of  millions 
of  relativities  must  therefore  be  neglected  to  bring 
the  case  of  the  motions  of  the  earth  down  to  a  work- 
able basis.  It  has  not  seemed  therefore  that  rela- 
tivities as  such  were  a  promising  line  of  attack.  It  has 
seemed  better  to  deal  with  the  most  essential  observed 
factors  after  the  Newtonian  method,  part  of  which 
have  a  relative  aspect  and  part,  such  as  mass,  inertia, 
and  energy,  an  inherent  aspect.  A  disproportionate 
stressing  of  relationships  has  been  one  of  the  sources 
of  error  all  down  the  ages.  The  relations  of  the  earth 
to  the  sun  were  the  same  as  now  in  early  historical 
times,  but  the  ancients  overstressed  what  was  most 
obvious  to  them  and  thought  the  earth  stationary  while 
they  made  Phoebus  drive  his  chariot  across  the  sky 
daily,  with  the  crystalline  sphere  following  at  night. 
Closer  study  showed  that  there  were  many  relativities 


xvi  A  Preliminary  Essay 

of  motion,  vast  spaces,  great  endowments  of  mass, 
inertia,  energy,  and  other  inherent  properties.  The 
recognition  of  these  led  to  modern  astronomy. 

In  a  complex  system  like  our  cosmos  motions  must 
of  course  be  relative.  The  relation  is  sometimes  causal 
and  sometimes  merely  incidental.  In  the  standard 
modes  of  thinking,  it  has  not  been  supposed  that  be- 
cause a  motion  is  relative  it  cannot  also  be  inherent 
or  absolute.  For  example,  the  earth  and  the  sun  re- 
volve about  their  common  center  of  gravity.  The 
curvature  of  their  paths  is  due  to  their  mutual  attrac- 
tions— their  relativity  if  you  please — but  if  the  mutual 
attraction  were  destroyed  or  neutralized,  both  sun  and 
earth  would  move  on  in  the  line  of  their  motions  at 
the  instant  mutual  gravity  ceased  to  cause  the  curva- 
ture. This  is  due  to  inertia,  an  inherent,  rather  than 
a  relative,  property.  The  destruction  of  the  relativity 
of  their  motions  would  not  destroy  their  absolute  or 
inherent  motions.  The  relativity  of  their  motions 
seems  to  be  only  a  modified  phase  of  the  absolutivity 
or  inherency  of  their  motions.  Right  here  lies  the  crux 
of  the  present  issue.  Einstein  stresses  relativity  to  the 
exclusion  of  the  absolute  or  inherent.  At  least  he  says 
specifically  that  there  is  no  absolute  motion.  It  may 
be  well  at  once  to  recognize  that  there  may  be  mis- 
understanding of  the  term  "absolute"  and  some  other 
terms,  for  the  usual  senses  of  terms  are  not  deferen- 
tially followed  by  the  relativists  and  the  non-Euclidian 
geometricians  whose  language  they  adopt.  This  group 
has  reached  preeminence  in  one  field  at  least;  they 
have  achieved  a  nomenclature  of  the  most  distinguished 
infelicity  attained  thus  far  in  the  history  of  linguistic 


A  Preliminary  Essay  xvii 

endeavor.  They  speak  of  dimensions  that  other  people 
merely  call  durations,  of  " fourth  dimensional  space," 
where  other  people  feel  able  to  think  of  only  three 
dimensions,  of  relative  motions  that  are  devoid  of 
absolute  motion,  and  so  on.  In  the  standard  way  of 
thinking  relativities  of  motion  are  phases  of  actual 
inherent  or  ''absolute"  motions.  As  the  inherent  are 
thought  the  more  basal,  they  have  been  given  pre- 
cedence and  the  relativities  fall  into  a  secondary  place. 
To  make  this  vital  point  clear,  let  everything  in  the 
universe  be  wiped  out  of  existence,  except  the  possi- 
bilities of  thought.  Then  let  there  be  introduced  a 
single  natural  unit,  any  unit  that  can  have  independent 
or  inherent  existence,  say  an  electron,  or  an  atom  of 
hydrogen,  or  a  quantum  of  energy.  This  unit  cannot 
itself  be  a  relation  or  a  relativity,  for  a  relation  implies 
at  least  two  things  that  are  related.  There  can  thus 
be  no  relativity  at  this  stage.  If  there  is  any  objection 
to  this,  will  Einstein  show  us  how  a  universe  could  be 
started  with  a  relativity? 

Let  a  second  unit  be  added  and  relativity  becomes 
possible.  Let  a  third  unit  be  added  and  the  possible 
relativities  rise  to  a  number  greater  than  the  absolutes. 
If  further  absolutes  are  added,  the  number  of  possible 
relativities  rises  to  even  greater  proportions.  In  a  com- 
plex system,  relativities  are  thus  likely  to  become  more 
numerous  than  absolutes.  They  are  likely  to  become  also 
more  apparent  because  one  sets  off  another.  But  instead 
of  the  relativities  putting  the  absolutes  out  of  existence, 
these  are  required  to  make  the  relativities  possible. 
To  the  naturalistic  thinker,  relativities  thus  serve  as 
a  form  of  proof  of  absolutivities.  The  naturalistic 


xviii  A  Preliminary  Essay 

thinker  does  not  see  how  relative  motions  can  exist, 
if  there  are  no  inherently  existent  things  to  move. 
He  does  not  see  why  a  motion  that  is  relative  may 
not  also  be  absolute.  Unless  the  relative  motions  when 
combined  algebraically  reduce  to  zero  it  seems  that 
there  must  be  some  absolute  motion  also.  The  think- 
ing public  would  like  to  know  if  Einstein  has  proved 
that  all  motions  would  become  nil  if  the  existing  rela- 
tions of  the  moving  bodies  were  blotted  out.  They 
would  like  to  have  the  privilege  of  scrutinizing  a 
putative  demonstration  that  there  is  no  absolute  motion. 

Further  than  this,  it  seems  impossible  for  Einstein 
to  deal  with  all  the  relativities  of  motion  of  the  earth 
or  of  any  other  body;  how  then  is  such  a  demonstra- 
tion possible?  It  is  easy  for  Einstein  to  show  the 
impossibility  of  dealing  with  the  absolute  in  a  full 
and  final  sense.  Is  it  not  equally  impossible  to  deal 
with  relativity  in  the  same  sense?  Especially  as  there 
may  be  more  relativities  than  absolutes? 

Let  us  turn  for  a  moment  to  some  tenets  that  pre- 
ceded the  Einstein  Theory  of  Relativity  and  led  up  to  it. 

First  comes  the  gloomy  forecast  of  Minkowski  that 
"From  henceforth  [1908]  space  in  itself  and  time  in 
itself  sink  to  mere  shadows  and  only  a  kind  of  union 
of  the  two  remains  independent."  The  layman  is 
puzzled  to  know  just  what  this  sinking  of  space  and 
time  into  mere  shadows  means,  as  also  just  what  the 
union  product  is,  and  why  the  union  has  independence 
when  its  constituents  have  none.  Usually  constituents 
are  more  independent  and  lasting  than  combinations. 
In  the  nomenclature  of  the  doctrine  of  which  this  is 
a  part,  time  is  styled  a  dimension  and  is  used  mathe- 


A  Preliminary  Essay  xix 

matically  as  a  variable  in  common  with  space-dimen- 
sions. A  "fourth  dimension"  and  "fourth-dimensional 
space"  carry  the  semblance  of  mystery  into  the  litera- 
ture of  the  doctrine.  Perhaps  it  will  help  toward 
clarity  to  re-state  Minkowski's  forecast  in  more  explicit 
terms :  From  henceforth  dimensions  in  themselves 
and  duration  in  itself  sink  to  mere  shadows  and  only 
a  kind  of  union  of  dimensions  and  duration  remains 
independent.  This  brings  the  forecast  within  the 
testing  power  of  the  public.  To  it  life  affords  no  more 
vivid,  sharply  defined  concepts  than  those  of  the  rooms 
and  appointments  of  home,  office,  or  shop.  Equally 
vivid  are  the  impressions  of  the  public  respecting  great 
architectural  creations.  All  these  are  combinations 
of  dimensions  and  spaces.  The  very  soul  of  architec- 
ture lies  in  adapting  occupied  space  (walls,  columns, 
etc.)  to  the  enclosure  of  empty  space  of  designed 
forms  and  dimensions  suited  to  the  special  purposes 
in  mind.  Now  has  it  seemed  to  designers  in  architec- 
tural planning,  or  to  builders  in  construction,  that  there 
is  any  such  inevitable  union  of  dimensions  with  time 
as  to  render  the  dimensions  shadowy  if  not  combined 
with  time?  Do  we  all  put  a  pinch  of  time  into  the 
dimensions  of  our  living  rooms  and  into  the  placing 
of  furniture  to  forestall  the  shadows  into  which  they 
otherwise  would  sink?  If  a  consensus  of  the  im- 
pressions of  the  best  thinking  people  of  the  world 
today,  in  such  matters — including  expert  designers  and 
builders — were  taken,  would  it  not  disclose  the  view 
that  the  senses  of  dimensions,  irrespective  of  time, 
and  of  time,  irrespective  of  dimensions,  have  each 
grown  more  distinct  and  precise  as  experience  has 


xx  A  Preliminary  Essay 

increased  intelligence  and  capacity  in  these  lines?  It 
certainly  puzzles  the  layman  to  form  a  clear-cut  im- 
pression of  how  and  why  the  concepts  of  space  and 
time  sink  into  obscurity  and  lose  their  independence 
if  studied  independently.  It  is  equally  puzzling  to 
picture  the  precise  nature  of  the  union  of  time  and 
space  that  is  urged  to  take  their  individual  places.  If 
one  looks  to  this  postulated  union  for  some  inspiring 
esthetic  effect  to  take  the  place  of  the  old  concepts 
foredoomed  to  sink  into  shadows,  do  any  high  lights 
stand  forth?  If  a  persistent  student  tries  to  follow 
the  line  of  the  new  thought  back  to  its  source,  is  he 
likely  to  land  elsewhere  than  in  the  infinitesimal,  or 
the  inaccessible,  or  the  indeterminable? 

To  an  investigator  of  experience,  a  feeling  of  in- 
security in  applying  results  arises,  if  time  is  put  into 
the  equations  simply  as  a  variable  and  is  handled  merely 
as  a  variable.  The  loss  of  any  distinguishing  mark 
in  the  computative  process  removes  the  result  one  step 
away  from  intimate  dependence  on  the  concrete,  and 
one  step  toward  that  freedom  in  mathematical  hand- 
ling which  comes  of  the  absence  of  bondages  to  the 
actual.  Every  loss  of  such  bondage  to  special  fact 
unfetters  the  mathematical  processes  and  gives  them 
freer  sweep  and  greater  scope.  From  that  point  of  view 
there  is  a  gain,  but  it  endangers  the  specific  application 
of  the  results  to  the  actual  case.  The  thrifty  workman 
multiplies  five  days'  work  by  four  dollars  a  day  in 
sheer  disregard  of  mathematical  propriety;  but  he 
thereby  holds  fast  to  the  facts  of  his  toilsome  lot.  He 
would  gladly  multiply  four  days'  work  by  five  dollars 
a  day,  but  his  employer  objects.  Stripped  of  trouble- 


A  Preliminary  Essay  xxi 

some  realities,  five  times  four  are  twenty;  the  problem 
is  easy  and  the  process  elegant.  Three  hundred  thou- 
sand kilometers  per  second  is  mathematically  much  the 
same  as  300,000  seconds  per  kilometer.  If  the  labels 
are  not  religiously  glued  to  the  factors,  the  result  is 
liable  to  become  a  composite  product  that  cannot  be 
readily  unscrambled.  Mathematics  is  undoubtedly  the 
greatest  achievement  thus  far  attained  by  man  through 
purely  intellectual  processes;  it  is  to  be  held  in  high 
reverence  on  this  account,  but  it  is  not  to  be  forgotten 
that  many  of  its  great  achievements  have  been  attained 
by  generalizations  and  abstractions  that  gave  the  mathe- 
matical processes  the  freest  possible  sweep.  Great  as 
are  its  virtues,  it  has  not  wholly  escaped  the  vices  of 
its  virtues.  Results  reached  by  means  of  the  abstract 
and  the  imaginary  are  likely  to  carry  these  qualities 
into  the  product.  To  secure  concrete  results  the  pro- 
cesses must  be  fettered  by  severe  bondage  to  the  facts, 
however  trammeling  these  may  be.  And  so  the  thrifty 
workman  is  true  to  the  admonitions  of  experience  in 
clinging  fast  to  his  method  of  multiplying  days  of 
work  by  the  wage  per  day,  for  these  are  to  him  the  hard 
realities. 

FitzGerald  and  his  followers,  to  meet  the  dilemma 
offered  by  the  negative  outcome  of  the  famous 
Michelson-Morley  experiments,  advanced  the  view  that 
the  dimensions  of  all  measuring  rods  vary  not  only 
according  to  the  velocity  of  their  motions  but  accord- 
ing as  they  move  endwise  or  sidewise,  while  all  time- 
keepers go  slower  as  they  are  carried  faster  through 
space.  Now  our  measuring-rods  form  the  chief  basis 
of  precision  in  physics  and  mechanics.  The  accuracy 


xxii  A  Preliminary  Essay 

of  time-keepers  is  equally  indispensable  in  astronomical 
work.  It  is  a  further  postulate  of  the  new  view  that 
these  variations  cannot  be  corrected.  Thus  about  all 
that  is  left  to  be  trusted  is  a  relativity  which  is  itself 
subject  to  change  and  is  presumably  always  changing. 
But  when  one  recalls  the  almost  infinite  plexus  of 
relativities  into  which  one  enters  when  he  follows 
actions  back  into  their  interminable  cosmic  connections, 
he  does  not  readily  see  how  even  such  postulated  var- 
iability can  really  be  demonstrated.  Are  these  postu- 
lates really  in  any  sense  demonstrations  or  are  they 
merely  weird  speculations  devised  to  escape  the  dilemma 
of  a  disappointing  experiment  ? 

A  further  step  toward  the  Einstein  theory  of  rela- 
tivity was  the  electron  theory  of  Lorentz.  There  is 
no  question  about  the  importance  of  the  new  revela- 
tions regarding  the  constitution  of  matter,  nor  any 
doubt  that  electro-magnetic  dynamics  play  an  impor- 
tant function  in  cosmic  affairs,  but  all  this  is  quite 
apart  from  the  question  whether  these  are  to  be 
interpreted  as  relativities  or  not.  The  electron  is  not 
typical  neutral  matter  such  as  forms  the  basis  of  the 
Newtonian  mechanics ;  it  is  merely  one  of  the  elements 
of  such  matter.  The  electron  always  appears  to  be 
derived  from  neutral  matter  or  from  matter  negatively 
charged.  Its  strangest  feature  is  that  it  appears  to 
develop — or  else  to  attach  to  itself — increased  mass 
as  its  velocity  is  increased.  This,  taken  by  itself,  may 
easily  be  thought  to  require  a  radical  change  in  standard 
ideas  of  mass.  But  in  the  neutral  state  of  matter,  the 
electron  is  mated  with  an  equivalent  electrical  charge 
of  the  opposite  type.  This  opposite  charge  has  never 


A  Preliminary  Essay          xxiii 

been  separated  from  matter  and  may  merely  be  matter 
deficient  in  electric  energy,  as  held  by  Franklin.  If 
this  is  the  correct  view,  the  law  that  action  and  re- 
action are  equal  and  in  opposite  directions  seems  to 
make  it  quite  sure  that  when  an  electron  is  shot  forth 
with  its  extraordinary  endowment  of  energy,  an  equiv- 
alent deficiency  of  energy  is  left  behind  in  the  form  of 
the  opposite  electric  state.  In  a  summation  of  effects 
on  a  large  scale,  as  in  astronomical  problems,  the 
balance  between  this  extraordinary  endowment  of  the 
electron  and  this  equally  extraordinary  deficiency  left 
behind,  should  give  a  result  of  the  order  of  neutral 
matter,  the  basis  of  the  Newtonian  mechanics.  As 
neutral  matter  overwhelmingly  surpasses  charged  mat- 
ter in  the  earth  and  as  the  increments  and  decrements 
of  charged  matter  offset  one  another,  the  sum  total 
of  differences  between  electro-magnetic  and  ordinary 
mechanics  largely  disappears  so  far  as  astronomical 
problems  and  most  ordinary  problems  are  concerned. 
Before  any  material  difference  can  be  claimed,  it  is 
necessary  to  show  that  the  earth  as  a  whole  is  per- 
sistently charged  either  positively  or  negatively,  and 
that  the  charge  has  a  value  large  enough  to  be  material 
when  compared  with  the  mass  of  the  earth.  Thus  far, 
any  such  appreciable  charge  has  seemed  improbable; 
certainly  it  has  not  been  shown  to  be  important  in 
astronomical  calculations.  But  it  is  of  course  a  matter 
that  invites  investigation,  and  may  yet  prove  to  have 
appreciable  importance.  But  even  then  it  will  remain 
to  be  shown  that  it  confirms  the  claims  of  Einstein 
Relativity. 

The  influence  of  electric  and  magnetic  fields  upon 


xxiv          A  Preliminary  Essay 

bodies  moving  within  them  also  needs  recognition. 
When  such  influence  is  appreciable,  the  treatment  must 
of  course  follow  the  laws  of  electro-magnetic  dynamics 
rather  than  those  of  neutral  matter.  But  here  again 
it  must  be  recognized  that  there  are  positive  fields  of 
force  and  negative  fields  of  force  and  that,  in  any 
summation  for  a  body  like  the  earth,  it  is  the  algebraic 
sum  of  the  two  that  represents  the  general  effect.  In 
the  problems  of  great  masses  such  as  planets  and  stars, 
it  does  not  appear  from  present  evidence  that  electric 
and  magnetic  fields  increase  or  diminish  appreciably 
the  value  of  the  results  given  by  Newtonian  mechanics. 
In  the  study  of  very  diffuse  bodies,  such  as  the  comas 
and  tails  of  comets,  and  the  diffuse  nebulas,  the  electric 
and  magnetic  states  very  likely  may  require  that  electro- 
magnetic mechanics  supplement  the  results  given  by 
ordinary  celestial  mechanics.  The  interchanges  of 
electrons  between  the  various  bodies  of  the  heavens 
may  have  value  enough  also  to  require  special  treatment 
in  addition  to  that  of  the  Newtonian  mechanics. 

But  until  these  several  modifying  influences  of 
electricity  and  magnetism  are  shown  to  have  value 
enough  to  require  a  modification  of  the  Newtonian 
mechanics,  the  proper  scientific  attitude  is  that  of 
reserve  with  an  attitude  hospitable  to  any  result  that 
may  be  supported  by  evidence  as  it  accumulates. 

All  these  phenomena  stand  on  their  own  evidences 
and  the  question  of  relativity  has  about  the  same  re- 
lationship to  them  that  it  has  to  the  phenomena  of 
neutral  matter. 

The  question  of  the  real  merit  of  the  Einstein 
contention  then  remains  to  be  settled  by  careful  and 


A  Preliminary  Essay 

critical  tests.  If  the  new  candidate  for  acceptance 
does  not  choose  to  enter  the  field  where  the  standard 
system  has  made  its  triumphs  well  known,  but  chooses 
its  own  tests  in  unfamiliar  fields,  the  nature  of  these 
fields  and  the  results  of  a  strict  application  of  the 
Newtonian  and  the  Einstein  systems  respectively  must 
be  set  forth  with  the  utmost  explicitness  and  fidelity 
to  give  the  public  a  fair  chance  to  pass  upon  the  merits 
of  the  case.  The  purpose  of  this  book  is  to  do  this 
for  the  two  astronomical  tests  to  which  appeal  has  been 
made.  The  reader  cannot  fail  to  be  impressed  with 
the  fullness,  care,  and  explicitness  of  Dr.  Poor's 
treatment. 

THOMAS  CHROWDER  CHAMBERLIN. 

THE  UNIVERSITY  OF  CHICAGO, 
August  30,  1922. 


CONTENTS 

CHAPTER  PACK 

AUTHOR'S  PREFACE iii 

PRELIMINARY  ESSAY vii 

I. — THE  THEORY  OF  RELATIVITY 3 

II. — THE     EVIDENCE     FOR     THE     RELATIVITY 

THEORY 52 

III. — THE  LAW  OF  GRAVITATION     .....      69 
IV. — THE  MOTIONS  OF  THE   PLANETS     .    .    .     108 

V. — THE    MOTION    OF    THE    PERIHELION    OF 

MERCURY 151 

VI. — THE  MOTIONS  OF  THE  PLANETS  AND  THE 

RELATIVITY  THEORY 185 

VII. — THE  ECLIPSE  PLATES  AND  THE  RELATIVITY 

THEORY 197 

VIII. — THE  OBSERVED  PHENOMENA  AND  CLASSICAL 

METHODS 227 

IX. — CONCLUSIONS 254 

APPENDICES 

I. — THE  MiCHELsoN-MoRLEY  EXPERIMENT  ON 

ETHER-DRIFT 261 

xxvii 


xxviii  Contents 

FACE 

II. — EINSTEIN  AND  THE  FIZEAU  EXPERIMENT    .  266 

III. — THE  MATHEMATICS  OF  RELATIVITY  .     .     .  268 

IV. — THE  DISPLACEMENT  OF  SOLAR  LINES  AND 

RELATIVITY 270 

INDEX 273 


TABLES 

TABLE  PAQB 

I. — PERTURBATIONS  OF  MERCURY  BY  VENUS     .  141 

II. — OBSERVED  DISCORDANCES  AND  THE  EINSTEIN 

MOTION 191 

III. — RADIAL  DISPLACEMENT  OF  INDIVIDUAL  STARS  218 

IV. — FINAL  DISCORDANCES  IN   THE  MOTIONS  OF 

THE  PLANETS 234 

V. — ANGULAR  DEPARTURES  FROM  RADIALITY  .     .  249 
VI. — COMPUTED  DEPARTURES  FROM  RADIALITY  .     .251 


XXIX 


ILLUSTRATIONS 
PLATES 

PLATE  PAGE 

ETHER  ROCK,  MOUNT  WILSON  OBSERVATORY, 
CALIFORNIA,  WITH  INTERFEROMETER 
HOUSE,  USED  BY  MILLER  IN  1921 

Frontispiece 

I. — ETHER-DRIFT  INTERFEROMETER  AS  USED  BY 

MORLEY  AND  MILLER  IN  1903-1905    ...       16 

II. — THE  SURFACE  OF  THE  SUN:  A  PHOTOGRAPH 
TAKEN  AT  THE  MOUNT  WILSON  OBSERV- 
ATORY   98 

III. — THE  ECLIPSE  OF  THE  SUN:  A  PHOTOGRAPH 

TAKEN  AT  SOBRAL,  BRAZIL,  MAY  29,  1919    212 

IV. — THE  SUN,  PHOTOGRAPHED  IN  THE  LIGHT  OF 
GLOWING  HYDROGEN  AT  THE  MOUNT 
WILSON  OBSERVATORY 230 

V. — THE  ETHER-DRIFT  INTERFEROMETER,  WITH 
CONCRETE  BASE,  AS  USED  BY  MILLER  IN 
1921  AT  ETHER  ROCK 264 


XXXI 


TEXT  CUTS  AND  FIGURES 

NUMBER  PACK 

i. — RETARDATION  EFFECTS  OF  A  CURRENT     .     .  12 

2. — THE  MlCHELSON-MORLEY  APPARATUS  ...  14 

3. — THE  RELATIVITY  OF  MOTION 21 

4. — ADDITION  OF  VELOCITIES 27 

5. — COORDINATES  AND  DISTANCE 39 

6. — DISPLACEMENT  OF  SPECTRAL  LINES    ....  58 

7. — THE  ROTATION  OF  MERCURY'S  ORBIT    ...  61 

8. — DEFLECTION  OF  LIGHT  RAYS  BY  THE  SUN     .  64 

9. — TRUE  AND  DEFLECTED  POSITIONS  OF  STARS    .  65 

10. — EPICYCLIC  MOTION 71 

ii. — KEPLER'S  LAWS  OF  PLANETARY  MOTION    .     .  78 

12. — THE  FALL  OF  THE  MOON  TOWARDS  THE  EARTH  82 

13. — MUTUAL    ATTRACTIONS     OF    PARTICLES    OF 

MATTER 85 

14. — ATTRACTIONS    OF    A    SPHERE    AND    OF    AN 

IRREGULAR  BODY 87 

15. — ATTRACTIONS  OF  SPHERES  AND  SPHEROIDS    .  89 

xxxiii 


xxxiv       Text  Cuts  and  Figures 

NUMBER  PAGE 

1 6. — FORCES  AND  TRAJECTORIES    ........  no 

17. — PROJECTILES  NEAR  THE  EARTH 112 

18. — ELLIPSES  OF  THE  SAME  SIZE 117 

19. — THE  ELEMENTS  OF  A  PLANET'S  ORBIT    .     .     .  121 

20. — MOTIONS  OF  A  SYSTEM  OF  BODIES     ....  125 

21. — COMPOSITION  OF  MOTIONS 134 

22. — TRANSIT  LIMITS      .    . , 156 

23. — EXTERNAL  AND  INTERNAL  CONTACTS     .     .     .  157 

24. — A  HORIZONTAL  TELESCOPE 200 

25. — THE  ECLIPSE  FIELD 203 

26. — INDIVIDUAL  RESULTS  OF  ECLIPSE  AND  COM- 
PARISON PLATES 215 

27. — THE  DISCORDANT  RESULTS:  STARS  5  AND  n  217 

28. — FAILURE  OF  REFRACTION  FORMULAS  ....  243 

29. — COMPARISON  OF  THE  EINSTEIN  AND  THE  RE- 
FRACTION EFFECTS 252 


Gravitation  versus  Relativity 


Gravitation  versus  Relativity 

CHAPTER  I 

THE  THEORY  OF   RELATIVITY 

THE  THEORY  OF  RELATIVITY,  as  developed  by  Pro- 
fessor Albert  Einstein,  is  an  attempt  to  explain  the  ap- 
parent results  of  certain  intricate  optical  experiments  by 
a  complete  reconstruction  of  our  fundamental  ideas  in 
regard  to  space  and  time.  From  the  earliest  days  of 
scientific  thought,  time  and  space  have  been  considered 
as  independent :  time  flowing  on  uniformly  regardless 
of  the  countless  bodies  in  the  universe  and  of  their 
motions  to  and  fro  through  endless  and  limitless  space. 
An  interval  of  time  has  always  been  regarded  as  the 
same  under  all  conditions  and  throughout  all  space ;  an 
hour  as  identically  the  same  for  a  person  at  rest  and 
for  an  aviator  flying  at  one  hundred  miles  per  hour, 
for  a  man  in  New  York  City,  for  an  hypothetical  in- 
habitant of  Mars,  and  at  the  most  distant  star.  But 
with  Einstein  all  this  is  changed;  space  and  time  are 
bound  together,  space  cannot  exist  without  time,  and 
time  changes  with  space  and  with  the  motions  of  the 

3 


4       Gravitation  versus  Relativity 

material  bodies  therein.  According  to  the  theory  of 
relativity  the  interval  of  time,  known  as  an  hour,  varies 
from  place  to  placs,  jt  is  different  for  the  person  at  rest 
and  for  the  aviator;  it  would  appear  longer  for  an 
*Sl£$tiC^^  shorter  to  an  observer  on 

the  slow  moving  Neptune. 

Motion  is  the  basis  of  our  conception  of  space.  We 
move  freely  about  on  the  earth  and,  as  we  move,  we  en- 
counter many  objects;  some  freely  moving  and  some 
apparently  fixed.  We  soon  learn,  however,  that  rest 
and  motion  are  relative  terms;  that  objects,  which 
from  one  point  of  view  may  be  considered  as  fixed, 
are  really  in  motion.  To  a  passenger  on  a  steamer  the 
decks,  the  cabins,  the  port-holes  are  fixed;  but  the 
steamer  is  in  motion  and  these  relatively  fixed  objects 
are  being  carried  from  port  to  port. 

Further  the  earth  itself  is  in  rapid  motion ;  it  rotates 
on  its  axis  and  it  sweeps  through  space  in  a  great 
curved  path  about  the  sun.  And  other  bodies,  similar 
to  the  earth,  are  found  participating  in  like  motions 
about  the  sun ;  Mars,  Jupiter,  Saturn,  and  hundreds  of 
minor  planetoids  and  comets.  These,  with  the  earth, 
form  the  Solar  System,  and  their  motions  relative  to 
the  sun  and  to  one  another  can  be  measured  and  their 
paths  determined.  But  even  the  Solar  System  is  not 
at  rest ;  delicate  and  long  continued  investigations  show 
that  the  sun  and  all  its  attendant  planets  are  moving 
among  the  stars.  The  stars  themselves  are  in  motion. 


The  Theory  of  Relativity          5 

All  these  moving  objects,  the  earth,  the  planets,  the 
stars  must  be  in  something,  and  that  something,  in 
which  they  exist  and  have  their  being,  is  what  we  call 
space. 

Thus  from  our  every  day  experience,  confirmed  by 
an  endless  series  of  measurements,  is  derived  our  fun- 
damental concept  of  space — a  general  receptacle  in 
which  things  have  their  existence.  As  all  bodies,  with 
which  we  come  into  contact,  have  three  dimensions, 
length,  breadth,  and  thickness,  so  the  general  receptacle 
must  have  three  dimensions  and  must  extend  in  all 
directions  beyond  the  farthest  known  or  imaginable 
thing.  The  general  container,  space,  contains  every- 
thing; it  is  itself,  however,  devoid  of  all  material  attri- 
butes; it  is  boundless,  limitless.  Each  point  of  space 
is  like  each  and  every  other  point;  each  point  has  one 
attribute  and  one  attribute  only,  that  of  position. 

Material  things  move  about,  or  are  moved,  in  this 
general  receptacle.  Certain  objects  are  noted  as  mov- 
ing in  cycles,  appearing  now  in  one  position,  now  in 
another,  and  then  back  again  to  the  first  position;  a 
flash  of  light  appears,  disappears,  only  to  reappear 
again  in  the  same  place.  To  such  recurrent  phenomena 
is  due  the  concept  of  time.  Although  each  flash  ap- 
pears at  the  same  point,  yet  each  flash  is  different. 
Something  has  happened,  and  we  need  a  system,  a 
something  to  distinguish  the  one  flash  from  the  other. 
Time  is  such  a  system,  it  is  the  system  by  which  we  con- 


6       Gravitation  versus  Relativity 

nect  together  events  as  they  actually  happen;  it  is  the 
wire,  so  to  speak,  on  which  we  string  the  beads  of 
successive  events.  In  our  concept  we  separate  the  sys- 
tem from  the  material  bodies,  and  conceive  of  absolute, 
true  time  as  flowing  at  a  constant  rate,  unaffected  by 
material  things,  or  by  the  motions  of  material  bodies. 
We  conceive  of  time  as  being  the  same  everywhere 
and  under  all  conditions.  The  instant,  known  as  5 
o'clock  in  the  afternoon  of  June  loth  in  New  York 
City,  is  a  definite  instant  everywhere  throughout  space ; 
the  same  in  New  York,  in  London,  on  Mars,  on  Jupiter, 
and  on  Sirius :  a  minute,  an  hour,  a  day,  each  measures 
identically  the  same  interval  of  time  at  every  point 
throughout  space. 

There  is,  however,  an  essential  difference  between 
space  and  time,  or  rather  between  our  fundamental  con- 
cepts of  these.  Space  is  reversible,  time  is  not.  One  can 
freely  pass  from  point  to  point  in  space  and  back  again ; 
within  limits  space  is  under  our  control,  we  can  occupy 
a  definite  point  or  not  at  our  pleasure.  Not  so  with 
time,  it  flows  uniformly  in  one  direction,  we  pass  for- 
ward but  not  backward,  and  the  speed  of  the  passage 
is  beyond  all  control.  We  must  pass  forward  in  time, 
whether  we  choose  or  not,  and  all  beings  and  all  things 
pass  forward,  in  time,  at  the  same  rate.  One  can  travel 
from  New  York  to  Boston  and  back  again  at  pleasure ; 
but  one  cannot  pass  from  to-day  to  yesterday,  nor  from 
yesterday  to  to-morrow :  the  days  pass  and  time  rolls 


The  Theory  of  Relativity  7 

on  without  check  or  hinderance,  and  is  the  same  for 
every  being  and  for  every  material  body  in  the  universe. 

Space  is  three  dimensional  and  reversible;  time  is 
of  one  dimension  and  irreversible.  Both  time  and 
space  exist  independently,  and  independent  of  any  mate- 
rial thing,  or  body. 

There  is  a  third  concept,  not  so  fundamental  as  those 
of  time  and  space,  but  still  essential  to  our  ideas — the 
concept  of  the  ether.  The  necessity  for  the  ether  arises 
from  modern  experiments  with  light.  Light  is  emitted 
by  a  candle,  by  the  sun,  by  a  distant  star,  and  some  how, 
in  some  way,  is  transmitted  through  space  to  the  eye  of 
an  observer.  Sir  Isaac  Newton  thought  of  this  trans- 
mission as  the  actual  passage  through  space  of  minute 
particles  of  matter;  he  thought  a  luminous  body  shot 
forth,  in  all  directions  and  at  immense  speed,  continu- 
ous streams  of  extremely  minute  particles,  or  cor- 
puscles. To  him  space  could  be  an  empty  void,  in 
which  moved  the  material  bodies  of  the  universe  and 
through  which  were  shot  the  minute  corpuscles  of 
light.  This  theory  explained  all  the  phenomena  known 
at  the  time  of  Newton,  but  it  cannot  explain  phenomena 
known  to-day,  nor  experiments  which  can  be  made 
with  simple  apparatus.  It  is  now  known  that  no  actual 
transmission  of  matter  takes  place  when  light  passes 
from  a  candle  to  one's  eye ;  light  is  now  known  to  be  a 
mode  of  motion,  a  type  of  wave  motion  analogous,  in 
a  remote  degree,  to  the  waves  of  the  ocean.  These 


8       Gravitation  versus  Relativity 

water  waves  travel  along  the  surface  of  the  ocean,  but 
the  water  itself  remains  practically  at  rest;  each  little 
particle  of  water  rises  and  falls  with  the  wave,  and  then, 
after  the  wave  has  passed,  it  returns  to  its  original 
position.  The  waves  produced  by  a  hurricane  in  the 
southern  seas  travel  for  thousands  of  miles  to  break  at 
last  as  a  gentle  surf  upon  the  beaches  of  our  northern 
coasts,  but  the  warm  waters  of  the  southern  seas  re- 
main in  the  south,  and  the  surf  on  the  beach  is  formed 
of  the  icy  waters  of  the  north.  Just  as  there  cannot  be 
ocean  waves  without  water,  so  we  cannot  conceive  of 
light  waves  and  the  propagation  of  light  from  sun  to 
earth  and  from  star  to  star  without  an  interstellar  ocean 
or  medium.  And  this  medium,  which  we  cannot  see, 
nor  feel,  nor  weigh,  is  the  so-called  ether  of  the  physi- 
cist. It  is  the  ocean  which  fills  all  space  and  in  which 
float  the  earth,  the  sun,  the  planets,  and  the  unnumbered 
stars,  and  throughout  which  are  passing,  in  every  direc- 
tion, the  waves  of  energy,  which  we  know  as  light  and 
radiant  heat. 

In  this  conception  of  an  all-pervading  medium  the 
physicist  has  met  many  difficulties :  certain  phenomena 
apparently  necessitate  its  having  definite  properties  as 
to  elasticity  and  rigidity,  other  phenomena  apparently 
require  different  and  antagonistic  properties.  The  need 
of  such  a  medium  is  essential  to  clear  thinking,  but 
experiments  have  failed  to  determine  its  specific  quali- 
ties. One  property,  one  attribute,  however,  has  been 


The  Theory  of  Relativity  9 

considered  as  fully  determined — the  ether  has  been 
thought  to  be  at  rest,  to  be  as  a  whole  forever  station- 
ary. The  earth,  the  planets,  the  stars,  all  move  through 
this  ethereal  ocean  without  disturbance,  without  fric- 
tion. 

If  the  ether  be  at  rest,  then  light  and  light  waves 
should  furnish  the  physicist  with  a  very  delicate  way 
of  measuring  the  direction  and  speed  of  the  earth 
through  this  medium ;  the  ether  should  furnish  an  abso- 
lute standard  to  which  can  be  referred  all  the  varied 
motions  of  the  planets,  and  from  which  their  exact 
positions  and  motions  in  and  through  space  can  be  de- 
termined. Many  experiments,  therefore,  have  been 
made  to  detect  and  to  measure  the  motion  of  the  earth 
through  the  ether,  or  the  "ether-drift,"  as  it  is  called, 
but  without  definite  results.  The  most  famous  attempt 
was  the  Michelson-Morley  experiment  made  in  1887. 
As  this  experiment  is  the  basis  of  the  Relativity 
Theory,  the  principles  involved  in  it  should  be  thor- 
oughly understood.  Einstein,  himself,  refers  to  it  as 
"decisive." 

Light,  as  has  been  seen,  is  a  wave  motion  in  the 
ether;  a  disturbance  that,  once  set  up,  travels  through 
the  ether  in  all  directions  at  a  terrific  speed.  The  earth 
also  moves  through  the  ether,  but  at  a  very  sedate  pace 
as  compared  to  light  waves.  If  the  earth  and  a  ray 
of  light  be  travelling  in  the  same  direction,  then  the 
ray  will  overtake  the  earth  and  pass  it,  but  it  should 


io     Gravitation  versus  Relativity 

pass  more  slowly  than  if  the  earth  were  at  rest.  Again, 
if  the  ray  and  the  earth  be  travelling  in  opposite  direc- 
tions, the  two  should  pass  at  a  much  greater  relative 
speed.  It  is  the  case,  if  you  like,  of  two  automobiles 
on  a  country  road:  if  they  are  both  travelling  in  the 
same  direction,  the  faster  car  gradually  overhauls  and 
passes  the  slower;  while,  if  they  are  travelling  in  oppo- 
site directions,  they  flash  by  in  a  moment.  Now,  the 
speed  of  light  is  so  enormous  as  compared  to  that  of 
the  earth,  one  hundred  and  eighty-six  thousand 
(186,000)  miles  per  second  as  against  only  nineteen 
(19)  miles,  that  it  is  impossible  to  measure  directly 
the  relative  speeds  of  a  ray,  when  passing  and  when 
overtaking  the  earth.  When  passing,  the  relative  speed 
would  be  186,019  miles,  when  overtaking,  185,981 
miles :  no  instrument,  no  method  yet  devised,  is  delicate 
enough  to  measure  the  speed  of  light  to  within  the 
38  miles  difference  between  these  quantities. 

What  cannot  be  done  directly,  however,  may  be 
done  indirectly.  Thus  Michelson  devised  a  method 
and  instrument  for  measuring,  not  the  actual  speeds  of 
the  two  rays  of  light,  but  the  ratio  of  their  speeds. 
By  an  ingenious  arrangement  of  mirrors  he  split  up 
a  ray  of  light  into  two  parts  and  sent  these  two  rays, 
or  parts  of  a  ray,  over  two  different  paths  of  the 
same  length,  the  two  paths  lying  in  different  directions 
relative  to  the  motion  of  the  earth.  Upon  the  com- 
pletion of  their  respective  journeys,  the  two  rays  re- 


The  Theory  of  Relativity          n 

turned  to  the  starting  point,  where  the  minute  interval 
between  the  times  of  their  respective  arrivals  was  ac- 
curately measured.  From  this  measured  difference  in 
the  times  required  for  the  rays  to  travel  the  same  dis- 
tance, the  difference  in  their  respective  speeds  can,  of 
course,  be  found  by  simple  calculation. 

A  simple  illustration  will  make  the  underlying  prin- 
ciple of  this  experiment  clear :  an  illustration  that  has 
been  used  many  times.  But  first,  it  will  simplify  mat- 
ters perhaps,  if  we  consider  the  earth  at  rest  and  the 
ether  as  drifting  by.  The  motion  of  the  two — the  earth 
and  the  ether — is  relative,  and  it  makes  no  difference 
to  which  we  attribute  the  motion.  Consider  a  motor- 
boat,  capable  of  running  10  miles  per  hour  in  still 
water,  making  regular  trips  up  and  down,  or  across,  a 
river  in  which  there  is  a  steady  current  of  two  miles 
per  hour.  On  a  trip  up  the  river  against  the  current 
the  boat  will  make  a  net  gain,  by  the  shore,  of  8  miles 
per  hour;  running  down  the  stream  the  boat  will  pass 
the  shore  at  the  rate  of  12  miles  per  hour.  It  would 
take  the  boat  30  minutes  to  go  up  the  stream  4  miles, 
but  only  20  minutes  for  the  return  trip  with  the  cur- 
rent; a  total  of  50  minutes  for  the  whole  return  trip 
of  8  miles.  In  still  water,  however,  the  boat,  travelling 
at  the  rate  of  a  mile  every  six  minutes,  could  make  an 
8  mile  trip  in  48  minutes.  In  other  words,  although  the 
boat  travels  the  same  distance  with  and  against  a  river 
current,  yet  the  effect  of  the  current  is  to  retard  the 


12     Gravitation  versus  Relativity 

boat,  to  lengthen  the  time  required  to  make  a  complete 
round  trip. 

A  moment's  consideration  will  show  that  a  similar, 
though  smaller,  retardation  takes  place  when  the  trip 


&&  -4SHJJ      '  -  ;2T 

Fig.  i.    Retardation  Effects  of  a  Current. 

is  made  across  the  current  in  any  direction.  If  the  boat 
be  headed  directly  across  the  stream,  it  will,  at  the  end 
of  24  minutes,  have  travelled  4  miles,  but,  during  this 
time,  the  current  will  have  carried  it  down  stream 
nearly  one  mile.  In  order  to  reach  its  destination  the 
boat  will  have  to  be  headed  up  stream  against  the  cur- 
rent, and  this  extra  mile  or  so  up  stream  will  take 
several  additional  minutes.  Such  a  course  unduly 
lengthens  the  trip;  the  quickest  trip  across  the  river 
can  be  made  by  heading  the  boat  slightly  up  river 


The  Theory  of  Relativity          13 

against  the  current,  just  enough  to  counteract  the  drift, 
and  by  keeping  the  boat  on  this  one  straight  course 
throughout  the  trip.  To  make  good  on  such  a  course 
the  boat  will  have  to  be  headed  up  river  about  a  com- 
pass point  and  the  trip  across  the  river  will  require  24 
minutes  and  some  29  seconds.  The  return  trip  will 
be  made  in  the  same  time,  so  that  the  entire  8  miles 
across  the  river  and  return  will  require  48  minutes 
and  some  59  seconds :  or  the  retardation  will  amount 
to  only  some  59  seconds,  as  against  2  minutes  for  the 
trip  up  and  down  stream. 

Thus  in  any  return  trip  the  traveller,  be  it  a  motor- 
boat  or  a  light  wave,  will  suffer  a  retardation  in  time, 
if  the  trip  be  made  in  a  moving  medium.  This  retarda- 
tion will  be  the  greatest  when  the  trip  is  directly  with 
and  against  the  current,  smallest  when  directly  across 
the  current.  Again  the  retardation  becomes  smaller 
and  smaller  as  the  speed  of  the  traveller  increases  in 
proportion  to  the  drift  of  the  medium.  In  the  case 
of  a  light  wave,  the  speed  of  the  wave  is  very  great 
in  comparison  with  the  speed  of  the  earth  through  the 
ether,  and  the  retardation  is,  therefore,  extremely  mi- 
nute. In  the  Michelson-Morley  experiment  the  return 
trip  was  but  a  few  yards  long,  and  the  calculated  re- 
tardation amounted  to  only  a  very  small  fraction  of  a 
billionth  of  a  second. 

To  measure  such  an  apparently  hopelessly  small  in- 
terval of  time  Michelson  devised  a  most  ingenious  in- 


14     Gravitation  versus  Relativity 

strument,  in  which  he  utilized  the  time  of  vibration  of 
a  light  wave  as  his  standard  measure  of  time.  Light 
waves  vibrate,  or  follow  one  another,  at  the  rate  of 
about  six  hundred  thousand  billion  a  second;  and  it 
was  this  interval  of  time  that  Michelson  used  to 
measure  the  relative  retardations  of  the  waves  travel- 
ling in  the  two  directions.  His  instrument,  therefore, 
was  capable  of  measuring  relative  retardations  many 
times  smaller  than  the  calculated  value.  The  optical 


Fig.  2.    The  Michelson-Morley  Apparatus 

principles  upon  which  this  measuring  device  is  based 
are  those  known  under  the  general  head  of  "interfer- 
ence." An  explanation  of  these  principles  and  of  the 
details  of  the  device  itself  would  be  out  of  place  here ; 


The  Theory  of  Relativity          15 

such  explanations  can  be  found  in  regular  text-books. 
It  is  only  necessary  to  state  that  the  device  is  founded 
on  sound  principles  and  is  capable  of  detecting  the 
minute  differences  required. 

It  is  different  however,  with  that  part  of  the  whole 
apparatus  which  contains  the  paths  of  the  rays  through 
the  ether.  This  must  be  clearly  and  fully  explained, 
for  upon  the  results  obtained  by  the  use  of  this  appara- 
tus depends  the  whole  theory  of  Relativity. 

The  apparatus,  made  of  steel,  was  in  the  form  of 
a  cross  which  floated  in  a  trough  of  mercury.  At  the 
end  of  each  arm  four  mirrors  were  placed  and  so  ar- 
ranged as  to  reflect  a  ray  of  light  back  and  forth  be- 
tween them.  A  ray,  starting  at  T  in  the  diagram, 
reached  a  mirror  at  M,  where  it  divided  into  two  parts. 
The  first  part  travelled  straight  on  to  the  mirror  i, 
where  it  was  reflected  back  to  a  corresponding  mirror 
at  the  other  end  of  the  arm ;  thence  it  was  reflected  back 
and  forth  from  end  to  end  of  the  arm  until  it  reached 
the  last  mirror.  Here  it  was  returned  over  its  course 
from  mirror  to  mirror  back  to  M  again,  where  it  was 
reflected  to  the  observer,  with  the  measuring  device,  at 
O.  The  second  part  of  the  ray  was  reflected  at  M  into 
the  transverse  arm  of  the  cross,  where  it  was  reflected 
back  and  forth  from  mirror  to  mirror  until  it  also 
reached  the  observer  at  O.  The  two  rays,  or  rather 
the  two  parts  of  the  same  ray,  thus  travelled  over  dif- 
ferent paths  at  right  angles  to  each  other  and  the  re- 


1 6     Gravitation  versus  Relativity 

tardation  of  the  one  ray  over  the  other  was  measured 
by  the  observer  with  his  delicate  interferometer. 

Of  course  is  is  absolutely  impossible  to  set  up  such 
an  instrument,  involving  the  placing  and  adjustment 
of  many  mirrors,  so  that  the  two  paths  are  exactly 
equal  in  length  and  so  that  each  and  every  mirror  is 
in  its  exact  alinement.  Nor  can  we  know  at  any  instant 
the  exact  direction  in  which  the  earth  is  plowing 
through  the  ether,  so  that  it  is  impossible  to  place  one 
arm  of  the  instrument  parallel  to  the  ether-drift  and  the 
other  at  right  angles  to  it.  In  any  one  fixed  position  of 
the  apparatus,  therefore,  an  observed  retardation  of  one 
ray  over  the  other  might  be  the  indication  merely 
of  instrumental  errors  of  adjustment,  errors  in  the 
lengths  of  the  arms,  in  the  alinement  of  the  mirrors, 
or  in  the  direction  of  the  instrument  as  a  whole.  But 
if  the  apparatus  be  rotated  so  that  the  arms  take  up 
various  positions  with  respect  to  the  drift,  then  the 
retardations  due  to  instrumental  errors  will  be  elimi- 
nated, and  that  due  to  the  drift  will  show  up. 

Such  observations  were  carried  on  over  a  long 
period  of  time,  at  different  hours  of  the  day,  at  dif- 
ferent seasons  of  the  year,  in  different  places  and  under 
different  conditions.  The  results  were  negative;  no 
constant  measurable  retardation  in  any  particular  direc- 
tion was  observed.  To  interpret  this  result,  return  for 
a  moment  to  our  motor-boat  illustration.  If  such  a 
boat  made  four  mile  return  trips,  up  and  down  stream, 


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The  Theory  of  Relativity          17 

across  the  stream,  and  in  any  and  every  direction  and 
every  trip  was  made  in  exactly  48  minutes,  then  one  and 
only  one  conclusion  could  be  drawn : — there  is  no  cur- 
rent, the  water  in  the  river  is  at  absolute  rest.  Similarly, 
on  the  face  of  the  Michelson-Morley  results,  light 
passes  an  observer  on  the  earth,  at  all  times  and  in  all 
directions,  at  the  same  speed;  the  earth  and  the  ether, 
if  indeed  there  be  an  ether,  move  together. 

The  results  of  the  Michelson-Morley  experiments, 
if  accepted  as  conclusive,  affect  the  physicist's  concep- 
tion of  a  motionless  ether  filling  all  space,  but  they 
do  not  necessarily  affect  in  any  way  the  fundamental 
concepts  of  time  and  space. 

After  many  fruitless  attempts  on  the  part  of  phy- 
sicists to  find  a  satisfactory  explanation  for  the  re- 
markable and  unexpected  results  of  this  experiment, 
Lorentz  suggested  the  contraction  theory,  known  under 
the  name  of  the  Lorentz-FitzGerald  theory.  Under 
this  theory  all  bodies  in  motion  are  compressed,  or 
shortened,  in  the  direction  of  the  motion.  The  amount 
of  this  compression  depends  upon  the  speed;  as  the 
speed  increases  the  contraction  becomes  greater  and 
greater.  A  steel  rod,  for  example,  may  measure 
exactly  one  yard  while  at  rest,  but,  when  set  in  motion 
in  the  direction  of  its  length  it  actually  becomes  shorter. 
A  mere  shift  in  direction  may  change  the  length  of 
the  rod,  for  the  rotation  of  the  earth  carries  all  bodies 
on  its  surface  in  an  eastward  direction  at  a  relatively 


18     Gravitation  versus  Relativity 

high  speed,  and  a  rod,  therefore,  under  this  theory  will 
actually  be  shorter  when  it  points  east  and  west  than 
when  north  and  south. 

Such  changes  in  length,  if  real,  cannot  be  directly 
measured,  for  all  bodies  contract  when  placed  in  mo- 
tion and,  therefore,  a  yard-stick  will  change  in  exactly 
the  same  way  and  in  exactly  the  same  proportion  as 
the  body  to  be  measured.  But  indirectly  through 
experiments  with  light,  as  in  the  Michelson-Morley 
experiments,  the  effect  of  the  contraction  becomes  ap- 
parent and  measurable.  Returning  to  our  illustration 
of  the  motor-boat  and  its  trips  in  the  river,  it  will  be 
recalled  that  it  required  about  one  minute  more  for  a 
return  trip  with  and  against  the  current,  than  for  a  re- 
turn trip  of  the  same  length  across  the  current.  But,  if 
now  the  up  and  down  trip  be  shortened  by  a  small 
fraction  of  a  mile,  leaving  the  across  current  trip  the 
original  length,  then  the  two  trips  might  be  made  in 
identically  the  same  interval  of  time.  So  with  the 
Michelson-Morley  experiment,  if  the  length  of  the 
arm  of  the  cross  automatically  becomes  shorter  when 
parallel  to  the  motion  of  the  earth  through  the  ether, 
then  the  time  interval  for  light  rays  travelling  in  this 
direction  will  be  reduced;  and,  if  the  shortening  be  in 
exactly  the  right  proportion,  then  the  two  intervals 
for  the  rays  along  the  two  arms  will  be  the  same,  and 
the  negative  result  of  the  experiment  fully  accounted 
for. 


The  Theory  of  Relativity          19 

The  actual  amount  of  contraction  required  to  ex- 
plain the  experiments  in  this  manner  is  very  minute, 
and  depends,  of  course,  upon  the  proportionate  speeds 
of  light  and  of  the  earth  through  the  ether.  At  the 
speed  with  which  the  earth  is  moving  in  its  orbit,  the 
necessary  contraction  in  the  length  of  a  body  is  less 
than  one  part  in  one  hundred  million :  about  one  inch 
in  the  distance  between  New  York  and  Denver. 

This  contraction  theory  of  Lorentz-FitzGerald  is 
a  possible  explanation  of  the  experiment;  it  is  one  of 
several  possible  explanations.  Einstein  offers  another 
solution  of  the  problem;  a  solution  of  a  radically  dif- 
ferent sort,  one  which  involves  the  fundamental  con- 
cepts of  time  and  space. 

Einstein,  as  the  very  basis  of  his  theories, 
assumes  that  the  Michelson-Morley  experiment  is 
"decisive" ;  that  "there  can  be  no  ether-drift,  nor  any 
experiment  with  which  to  demonstrate  it"  (63).*  He 
contends  in  substance  that  we  have  exhausted  all  means 
of  research  in  our  attempts  to  measure  absolute  motion 
through  space,  and  that,  having  failed,  it  is  because 
the  universe  is  so  constructed  as  to  make  it  impossible 

*  Relativity,  the  Special  and  General  Theory,  by  Albert  Einstein. 
Translated  by  R.  W.  Lawson,  1920,  page  63. 

The  followers  of  Einstein  do  not  always  agree  among  them- 
selves as  to  the  details  of  the  theory  and  as  to  the  meaning  of 
certain  principles  and  statements.  I  have,  therefore,  in  my 
attempt  to  explain  the  concepts  and  theories  of  Einstein,  con- 
fined myself  to  a  study  of  his  own  work.  The  many  references 
to  this  work  are  noted  in  the  text  by  the  number  of  the  page  on 
which  the  quotation,  or  statement,  is  to  be  found. 


20     Gravitation  versus  Relativity 

ever  to  detect  by  any  physical  experiment,  optical  or 
otherwise,  the  existence  of  absolute  motion,  or  motion 
through  any  ether,  or  other  medium,  which  may  per- 
vade all  space.  His  course  of  reasoning  appears  to  be, 
that,  as  it  has  been  "decisively"  shown  that  we  cannot 
measure  absolute  motion,  therefore  absolute  motion 
does  not  exist.  This  is  the  same  course  of  reasoning 
that  led  Congress  to  refuse  money  to  Langley  for  a 
continuation  of  his  experiments  with  flying  machines : 
— all  experiments  had  failed,  Langley's  machine  had 
fallen  into  the  Potomac,  and,  therefore,  a  successful 
flying-machine  was  an  utter  impossibility.  But,  con- 
gressional reasoning  to  the  contrary  notwithstanding, 
we  now  have  flying-machines ;  machines  built  upon  the 
very  principles  upon  which  Langley  was  working. 

Einstein  asserts  that  the  Michelson-Morley  experi- 
ment is  final  and  conclusive,  and  he  explains  the  result 
of  that  experiment  by  the  assertion  that  there  is  no 
"absolute  motion,"  there  is  no  "absolute  space,"  there 
is  no  "absolute  time."  All  motion  is  relative:  the 
steamer  moves  relatively  to  the  earth,  the  earth  moves 
relatively  to  the  sun,  the  sun  relatively  to  the  stars. 
Nothing  exists  independently  of  the  observer;  all  is 
relative,  nothing  is  absolute.  Hence  the  name: — 
RELATIVITY  THEORY. 

This  general  postulate,  or  assumption,  of  the  rela- 
tivity of  all  motion  and  the  non-existence  of  absolute 
motion  is  explained,  illustrated  and  enforced  by  Ein- 


The  Theory  of  Relativity          21 

stein  in  the  following  way.  He  supposes  two  observers, 
one  in  a  railroad  train  running  on  a  straight  stretch  of 
track  at  uniform  speed,  the  other  observer  standing 
beside  the  track  and  watching  the  train  go  by.  Just  as 
the  train  passes  the  watcher  on  the  ground,  the  person 
in  the  train  leans  out  of  the  window  and  drops  a  stone. 
This  stone  partakes  of  the  forward  motion  of  the  train, 
and  to  the  person  who  let  it  fall  it  appears  to  fall  in 
a  straight  line,  as  shown  in  the  accompanying  diagram. 
But  to  the  watcher  on  the  ground,  the  stone,  as  it  falls 
towards  the  earth,  appears  to  move  forward  in  the  same 
direction  as  the  train;  to  him  it  appears  to  describe 
a  curved  line,  a  parabola. 


Fig.  3.    The  Relativity  of  Motion. 


Which,  if  either,  is  the  true  path  of  the  stone;  the 
straight  line  as  it  appears  to  the  one  observer,  the 
parabola  as  the  other  sees  it,  or  is  it  some  other  curve, 
compounded  of  these  and  the  motion  of  the  earth? 
To  Einstein  the  answer  is  simplicity  itself,  the  stone 


22     Gravitation  versus  Relativity 

has  no  path.  "With  the  aid  of  this  example  it  is 
clearly  seen  that  there  is  no  such  thing  as  an  indepen- 
dently existing  trajectory  (lit.  "path-curve"),  but  only 
a  trajectory  relative  to  a  particular  body  of  refer- 
ence" (10). 

What  does  this  statement  of  Einstein  mean?  The 
stone  certainly  left  the  window  of  the  car  and  came  to 
rest  at  some  point  on  the  ground  at  the  side  of  the 
track.  How  did  it  get  from  the  window  to  the  ground  ? 
The  ordinary  common-sense  answer  would  be  that  the 
stone  travelled  in  some  curved  path  through  space, 
from  one  point  to  the  other.  It  is  true  that  this  path 
might  appear  differently  to  different  observers;  as  a 
straight  line  to  a  person  in  the  train,  as  a  parabola  to 
a  watcher  on  the  ground,  as  a  twisted  curve  to  an 
aviator  flying  diagonally  over  the  train :  but,  no  mat- 
ter how  the  path  appeared  to  these  various  observers, 
the  stone  travelled  in  one  single  definite  path ;  it  would 
have  travelled  the  same  path,  if  no  one  had  watched  it 
fall.  Now  this  simple  common-sense  statement,  that 
the  stone  actually  did  pass  from  the  one  point  to  the 
other  in  some  definite  path,  is  exactly  what  Einstein, 
in  his  statement  above  quoted,  denies.  He  states  that 
the  stone  had  no  path  independent  of  an  observer,  that 
the  path  it  travelled  depended  upon  the  person  who 
watched  it  fall,  that  it  actually  had  different  paths  for 
the  different  observers.  This  is  the  essence  of  relativ- 
ity :  the  path  that  the  stone  travels  is  a  joint  phenome- 


The  Theory  of  Relativity          23 

non  of  the  observer  and  of  the  stone.  In  the  absence  of 
either  the  observer  or  the  stone  there  would  be  no  path : 
the  path,  a  joint  phenomenon  of  the  observer  and  the 
stone,  exists  only  in  their  joint  presence. 

This  postulate  of  relativity  is  the  denial  of  the 
existence  of  any  reality  behind  our  observations.  The 
physical,  material  world  of  land  and  water,  of  trees 
and  houses,  of  men  and  women  does  not  actually  exist ; 
it  is  not  a  real  world.  It  exists  only  in  and  through 
the  observer,  and  is  different  for  different  observers. 
The  path  of  a  falling  stone  is  a  straight  line  for  one 
observer,  a  parabola  to  another;  a  steel  rod  is  a  yard 
long  to  one  person,  and  a  different  length  to  another; 
each  and  every  observer  is  correct;  the  stone  has  no 
"true"  path,  the  rod  has  no  "real"  length.  The  fan- 
tastic picture  of  the  cubist  is  as  true  to  nature  as  the 
work  of  a  Corot  or  a  Meissonnier. 

This  postulate  of  relativity  can  be  expressed  also  in 
mathematical  language;  in  terms  of  systems  of  co- 
ordinates, frames  of  reference,  and  of  transformation 
equations.  While  these  words  may  be  unfamiliar  to 
the  reader,  the  things  themselves  are  in  constant,  daily 
use.  The  charts  showing  the  fluctuations  of  the  stock 
market,  the  engineer's  diagrams  showing  the  relation 
between  speed  and  engine  power,  are  both  examples 
of  the  use  of  a  system  of  coordinates.  In  the  first  case 
the  horizontal  lengths  in  the  diagram  are  made  pro- 
portional to  the  time,  to  weeks,  months,  or  years,  and 


24     Gravitation  versus  Relativity 

the  vertical  lengths  proportional  to  the  price  of  stocks. 
Such  a  diagram  gives  at  once  the  price  of  the  stock 
on  any  date,  and  the  curve  connecting  the  various  prices 
shows  clearly  the  fluctuations  in  the  market  price.  The 
two  heavy,  fixed  boundary  lines  of  such  a  diagram,  one 
horizontal,  the  other  vertical,  are  technically  the  axes, 
and  the  lengths,  which  locate  any  point  on  the  price 
curve,  are  the  coordinates  of  that  point.  Any  number 
of  diagrams  may  be  made  to  represent  the  same  price 
fluctuations :  the  unit  of  time  may  be  the  week,  the 
month,  or  the  year ;  the  unit  of  price  may  be  the  dollar, 
the  pound,  or  the  franc.  In  changing  our  diagram 
from  one  price  unit  to  another,  from  the  dollar  to  the 
franc,  or  from  the  franc  to  the  pound,  we  must  have 
a  definite  relation  between  the  units,  we  must  know 
the  "rate  of  exchange."  This  rate  of  exchange  may 
remain  sensibly  the  same  for  long  periods  in  normal 
times,  or  it  may  suffer  violent  changes  from  day  to 
day,  as  in  the  abnormal  times  of  war. 

In  a  manner  entirely  similar,  the  mathematician 
specifies  the  position  of  a  point  in  space  by  referring 
it  to  three  mutually  perpendicular  planes  and  calls  such 
a  system,  a  system  of  coordinates  and  coordinate 
planes.  If  to  such  a  system  a  clock  be  added,  so  as  to 
fix  the  time  of  an  event  as  well  as  the  position  in  space 
at  which  it  occurs,  we  then  have  a  Frame  of  Reference; 
and  each  observer  of  physical  phenomena  is  supposed 
to  have  such  a  frame,  to  which  he  is  rigidly  attached 


The  Theory  of  Relativity          25 

and  with  which  he  moves.  To  return  to  the  example 
of  the  moving  train,  the  observer  who  drops  the  stone 
refers  its  motion  to  a  frame  rigidly  attached  to  the 
train  and  partaking  of  the  train's  motion;  the  watcher 
on  the  ground,  on  the  other  hand,  refers  the  motion 
of  the  stone  to  his  frame  of  reference  which  is  rigidly 
attached  to  the  earth.  The  diagrams,  or  equations, 
which  the  two  observers  draw  or  compute  to  repre- 
sent the  motions  of  the  falling  stone  will  be  different, 
but  from  one  diagram  we  ought  to  be  able  to  construct 
the  other,  provided  we  know  the  relative  motion  of  the 
two  frames,  provided,  in  other  words,  that  we  know 
the  rate  of  exchange.  In  technical  language,  to  pass 
from  one  reference  frame  to  another  we  need  "trans- 
formation equations." 

Now  the  observer  in  the  railroad  carriage  will 
formulate  laws  of  motion  and  of  falling  bodies  with 
reference  to  his  frame;  the  watcher  on  the  ground  will 
similarly  formulate  laws  in  reference  to  his  frame.  If 
the  laws,  thus  formulated  by  the  two  observers,  are 
fundamental  laws  of  nature,  they  should  be  identically 
the  same ;  the  two  systems  should  be  equivalent  for  the 
description  of  natural  phenomena.  When  the  observed 
phenomena  are  transferred  from  one  frame  of  refer- 
ence to  the  other  by  the  use  of  the  proper  transforma- 
tion equations,  then  the  fundamental  laws  derived  from 
the  observations  should  be  the  same  and  should  be 
expressed  in  identically  the  same  form.  According 


26     Gravitation  versus  Relativity 

to  the  relativitists,  all  laws  of  nature  can  or  should  be 
enunciated  in  such  forms  that  they  are  as  true,  in 
these  forms,  for  one  observer  as  for  another,  even 
though  the  observers,  with  their  frames  of  reference, 
be  in  uniform  motion,  without  rotation,  relative  to  one 
another.  According  to  Einsein,  when  K  and  K'  repre- 
sent two  different  coordinate  systems,  this  idea  is  ex- 
pressed in  the  following  explicit  terms:  "If,  relative 
to  K,  K'  is  a  uniformly  moving  co-ordinate  system 
devoid  of  rotation,  then  natural  phenomena  run  their 
course  with  respect  to  K'  according  to  exactly  the  same 
general  laws  as  with  respect  to  K.  This  statement  is 
called  the  principle  of  relativity  (in  the  restricted 
sense)."  (15). 

Now  the  principle  of  relativity  stated  in  this  mathe- 
matical form  is  equivalent  to,  or  tacitly  involves,  the 
denial  of  "absolute  rest" :  it  means  that  there  is,  in 
the  universe,  no  body,  no  thing,  completely  at  rest; 
that  there  is  no  motionless  ether.  For,  with  respect 
to  a  system  absolutely  at  rest,  with  respect  to  a  motion- 
less ether,  the  natural  laws  of  motion  are  capable  of 
being  formulated,  or  expressed,  in  a  particularly  simple 
and  unique  manner:  such  a  system,  therefore,  is  not 
equivalent  to  other  systems.  Thus  the  mathematical 
statement  of  the  relativity  principle  means  the  same 
thing  and  involves  the  same  ideas  as  the  statement  of 
the  principle,  heretofore  made,  in  general,  untechnical 
language. 


The  Theory  of  Relativity          27 

THE  SECOND  POSTULATE,  or  principle,  an- 
nounced by  Einstein,  states  that  the  velocity  of  light  in 
free  space  appears  the  same  to  all  observers,  regardless 
of  the  motions  of  the  source  of  light  and  of  the  ob- 
server. This  is  quite  different  from  the  old  idea,  or 
assumption,  that  the  velocity  of  light  in  space  is  uni- 
versally constant.  The  constant,  c,  for  the  velocity 
of  light  in  the  Einstein  formulas  is  not  the  actual 
velocity  of  the  wave  in  space,  in  the  ether,  but  refers 
to  the  observer's  measured  value  of  this  velocity  with 
respect  to  himself. 

In  order  to  understand  just  what  this  postulate 
means,  consider  again  the  case  of  the  motor-boat  in 
the  river.  The  current  is  running  down  stream,  from 
left  to  right,  with  a  speed  of  two  miles  per  hour,  and 
the  boat  is  capable  of  running  ten  miles  per  hour  in 
still  water.  If  a  floating  log  and  the  motor-boat  start 
side  by  side  in  the  stream  opposite  a  point,  A,  on  the 


Fig.  4.    Addition  of  Velocities. 

river  bank;  then  at  the  end  of  an  hour,  the  log  will 
have  floated  with  the  current  and  will  be  opposite  a 


28     Gravitation  versus  Relativity 

point,  B,  two  miles  below  A,  but  the  motor-boat,  run- 
ning at  full  speed,  will  have  been  carried  two  miles 
by  the  current  and  ten  miles  by  its  own  power,  or  will 
have  reached  a  point,  B',  twelve  miles  down  stream. 
In  other  words,  the  boat  will  have  passed  down  the 
river  by  a  distance  equal  to  the  sum  of  the  distances 
travelled  by  the  current  and  by  the  boat  itself.  This 
is  the  ordinary  theorem  of  the  addition  of  velocities. 
It  is  in  constant  daily  use,  in  one  way  or  another,  by 
practically  everyone;  it  is  taken  so  much  as  a  matter 
of  course  that  no  one  for  a  moment  doubts  its  truth 
and  validity. 

This  theorem  can  be  expressed  numerically  in  the 
form  of  an  equation.  Let  us  put  v  as  the  speed  of  the 
current,  and  w  as  that  of  the  motor-boat  in  still  water. 
Then  the  total  actual  speed  of  the  boat  down  the  river 
past  the  shore,  or  the  distance  covered  in  one  hour, 
will  be 

c  =  v  -f  w 

In  the  example,  v  was  equal  to  2  miles,  w  equal  to 
10  miles,  and  c,  the  actual  distance  down  stream  in  one 
hour,  was,  as  has  been  seen,  12  miles.  If  the  boat 
ran  up  stream  against  the  current,  then  v  would  be  sub- 
tracted from  w,  and  the  boat  would  pass  the  shore  at 
the  rate  of  8  miles  per  hour.  Now,  by  giving  v  and  w 
appropriate  values,  this  equation  can  be  made  to  apply 
to  any  case  that  may  occur,  to  streams  of  various 


The  Theory  of  Relativity          29 

velocities,  to  motor-boats  of  widely  different  speeds, 
to  railroad  trains  and  to  men  walking,  to  airplanes 
and  to  winds  of  various  strengths.  But  always,  under 
all  conditions,  for  all  velocities,  for  boats,  for  men, 
for  airplanes,  c  is  invariably  equal  to  the  sum  of  v 
and  w. 

Now  Einstein  applies  this  ordinary  theorem  of  our 
every-day  life  to  the  question  of  the  speed  of  light  as 
determined  by  two  observers,  one  in  a  railroad  train 
running  at  high  speed,  and  the  other  one  on  the  ground. 
In  this  application,  the  river  becomes  the  train  and 
the  motor-boat  becomes  the  wave  of  light.  The  speed 
of  the  train  is  v,  and  the  speed  of  light  relative  to  the 
train  is  w,  and,  therefore,  our  equation  gives  us  for 
the  speed  of  light  relative  to  the  ground, 

c  =  v  -f  w 

or,  the  velocity  of  light,  as  measured  by  the  observer 
on  the  ground,  should  be  greater  than  that  measured 
by  the  observer  in  the  train ;  it  should  in  fact  be  exactly 
equal  to  the  sum  of  the  speeds  of  the  train  and  of  light, 
as  measured  by  the  observer  in  the  train.  To  put 
this  another  way,  the  wave  of  light  should  pass  the 
observer  at  rest  on  the  earth  faster  than  it  could  over- 
take and  pass  the  train,  just  as  an  automobile  passes 
a  man  by  the  roadside  more  quickly  than  it  can  catch 
up  to  and  pass  a  car  running  in  the  same  direction. 
This  apparently  simple  and  common-sense  result  is, 


30     Gravitation  versus  Relativity 

however,  according  to  Einstein,  not  correct.  He  as- 
serts that,  in  the  case  of  light,  c  is  always  equal  to  w. 
Applied  to  a  motor-boat  and  a  running  stream,  this 
means  that  the  boat  would  appear  to  go  up  stream 
against  the  current,  or  across  the  current,  or  down 
stream  with  the  current  at  exactly  the  same  speed: 
applied  to  an  automobile,  that  it  appears  to  pass  the 
man  by  the  roadside  and  the  fast  moving  car  at  exactly 
the  same  rate.  Einstein  asserts  that  the  velocity  of 
light  appears  the  same  to  every  observer;  for,  accord- 
ing to  the  principle  of  relativity,  the  law  of  trans- 
mission of  light  must  be  the  same  for  the  railway 
carriage  as  for  the  ground.  Further,  the  "decisive'* 
experiment  of  Michelson  and  Morley  proved  this  to 
be  the  case;  the  velocity  of  light,  as  measured  by  them, 
was  always  the  same,  no  matter  what  the  motion  of 
their  apparatus. 

Of  course,  it  is  self  evident  that  so  long  as  the 
quantities  in  our  equation  have  their  common,  every- 
day significance,  c  and  w  cannot  be  equal,  unless  v  is 
zero.  But  Einstein  states  that  c  and  w  are  always 
equal,  no  matter  what  values  v  may  have.  Therefore, 
if  the  theory  of  relativity  be  true,  these  quantities, 
c,  v,  and  w,  must  differ  in  some  way  from  our  ordi- 
nary conception  of  them.  They  are  all  velocities ;  ratios 
of  a  distance  divided  by  a  definite  interval  of  time; 
feet  per  minute,  or  miles  per  hour,  for  example.  They, 
therefore,  involve  measures  of  both  distances  and  times. 


The  Theory  of  Relativity          31 

And  heretofore,  when  measuring  distances  and  times, 
we  have,  in  our  common-sense  way  of  looking  at 
things,  assumed  that : 

1.  The  distance  between  two  points  of  a  rigid 

body  is  independent  of  the  condition  of 
motion  of  the  body  of  reference :  that  is,  a 
yard-stick  is  a  yard  long  whether  on  the 
ground  or  in  the  train. 

2.  The  time  interval  between  two  events  is  in- 

dependent of  the  condition  of  motion  of 
the  body  of  reference :  that  is,  a  second  of 
time  for  an  observer  on  the  ground  is 
exactly  the  same  interval  as  a  second  of 
time  for  the  observer  in  the  train. 

These  assumptions,  which  seem  perfectly  natural  and 
in  accord  with  our  every-day  experience,  are,  accord- 
ing to  Einstein,  "unjustifiable"  (36),  and,  if  discarded, 
then  the  theorem  of  the  addition  of  velocities  becomes 
invalid,  c  and  w  may  always  be  equal,  and  the  Theory 
of  Relativity  emerges  triumphant. 

This  statement  of  Einstein  means  one  thing  and 
one  thing  only :  that  a  rigid  body  appears  to  be  of  dif- 
ferent lengths,  when  in  motion  and  when  at  rest;  that 
an  interval  of  time  appears  to  be  different  to  an  ob- 
server at  rest  from  what  it  does  to  an  observer  in 
motion.  The  mathematical  part  of  the  problem  is  to 
find  some  definite  relation  between  lengths  and  inter- 
vals of  time,  as  measured  by  observers  at  rest  and  in 


32     Gravitation  versus  Relativity 

motion,  such  that  the  measured  velocity  of  the  ray  of 
light  shall  always  be  the  same,  such  that  c  shall  always 
be  equal  to  w  in  our  equation.  This  condition  of 
equality,  essential  to  the  relativity  theory,  leads  to  a 
perfectly  definite  relation,  to  specific  transformation 
equations  or,  in  banking  parlance,  to  a  fixed  "rate  of 
exchange."  These  equations  are  known  as  the 
"Lorentz  Transformation  Equations,"  and  they  involve 
the  velocity  of  the  moving  body  and  the  velocity  of 
light. 

In  accordance  with  these  equations,  the  length,  a, 
of  a  rigid  body  in  motion  with  a  velocity  v  is  altered 
so  as  to  become  : 


a  X 

where  c  is  the  velocity  of  light.  That  is,  according  to 
Einstein,  "The  rigid  rod  is  thus  shorter  when  in 
motion  than  when  at  rest,  and  the  more  quickly  it 
is  moving,  the  shorter  is  the  rod"  (42).  In  simple  lan- 
guage, this  statement  of  Einstein  means  that  a  yard- 
stick is  shorter  when  it  is  placed  lengthwise  on  the 
floor  of  a  car  travelling  forty  miles  per  hour,  than 
when  at  rest  on  the  ground ;  that  it  becomes  still  shorter 
when  carried  by  an  airplane  at  one  hundred  and  fifty 
miles  an  hour.  It  means  that,  if  such  a  yard-stick  be 
pivoted  on  the  wing  of  an  airplane  in  flight,  it  auto- 
matically increases  and  decreases  in  length,  as  it  is 
placed  parallel  to  the  direction  of  flight,  or  cross-wise 


The  Theory  of  Relativity          33 

on  the  wing.  Further,  from  this  equation  it  would 
appear  that,  if  a  material  body  could  be  given  the 
velocity  of  light,  its  length  would  become  zero;  for  in 
this  case  the  fraction  v/c  would  become  c/c,  or  unity, 
and  the  quantity  under  the  square  root  sign  would  thus 
become  zero.  The  velocity  of  light,  c,  is  thus  a  limit- 
ing velocity,  which  can  never  be  exceeded. 

Now  although  this  assertion,  or  assumption,  of 
Einstein  may,  at  first,  appear  strange  and  contrary  to 
common-sense,  yet  it  does  not  directly  conflict  with 
our  old  fundamental  concepts  of  time  and  space. 
Bodies  contract  under  pressure,  they  expand  with 
heat,  and  such  expansion  and  contraction  is  perfectly 
understandable  and  in  accord  with  the  concept  of 
space.  It  does  not  require  any  very  radical  change 
in  our  fundamental  concepts,  or  methods  of  thinking, 
to  conceive  of  motion  as  acting  upon  a  body  as  a  sort 
of  pressure  and  of  making  it  actually  smaller.  This 
is  the  Lorentz-FitzGerald  contraction  theory,  as  here- 
tofore explained. 

Further,  as  to  time  and  intervals  of  time,  the 
Lorentz  transformation  equations  show  that  a  time 
interval,  t,  changes  for  a  body  in  motion  and  becomes : 


The  larger  the  velocity,  v,  of  the  body,  the  smaller 
the  denominator  of  this  fraction  and  the  greater  the 


34     Gravitation  versus  Relativity 

time  interval.  The  time  interval  being  larger,  a  clock 
must  run  more  slowly,  or  as  Einstein  puts  it:  "As  a 
consequence  of  its  motion  the  clock  goes  more  slowly 
than  when  at  rest"  (44) . 

Now  this  statement,  or  assertion,  or  assumption  of 
Einstein  in  regard  to  time  and  time  intervals  is  in 
direct  opposition  to  our  fundamental  concepts,  it  vio- 
lates our  whole  mode  and  method  of  thinking.  Here- 
tofore, time  has  been  thought  of  as  being  independent 
of  everyone  and  everything:  time  was  the  same  for 
all  portions  of  space,  for  all  bodies,  whether  in  motion 
or  at  rest :  a  minute  was  a  minute  the  world  around 
and  everywhere  in  space.  This  identity  of  time  and 
time  intervals  Einstein  denies:  according  to  his  rela- 
tivity theory,  time  depends  upon  motion,  and  every 
body  has  its  own  particular  time :  "unless  we  are  told 
the  reference-body  to  which  the  statement  of  time 
refers,  there  is  no  meaning  in  a  statement  of  the  time 
of  an  event"  (32).  The  faster  the  body  moves,  the 
longer  become  the  time  intervals.  The  earth  travels 
about  the  sun  at  a  rate  of  19  miles  per  second,  Mercury 
at  from  23  to  35  miles  per  second  depending  upon  its 
place  in  its  orbit,  and  Neptune  at  only  $]/$  miles  per 
second.  To  an  observer  on  Neptune,  therefore,  the 
interval  of  time,  which  we  know  as  a  year,  would 
appear  shorter,  while  to  an  astronomer  on  Mercury  the 
year  would  appear  longer.  While  the  speeds  of  the 
planets  thus  differ  greatly,  yet  they  are  all  very  small 


The  Theory  of  Relativity         35 

fractions  of  the  speed  of  light,  and  hence  the  variations 
in  time  intervals  will  be  very  minute,  only  about  one 
part  in  a  hundred  million.  The  lengths  of  the  year, 
as  measured  on  the  earth  and  on  Mercury,  would  differ 
by  only  a  couple  of  seconds  or  so. 

This  basic  idea  of  the  relativity  of  time  should  be 
thoroughly  understood,  and  a  further  illustration  may 
aid  one  in  forming  a  conception  as  to  what  relativity, 
as  enunciated  by  Einstein,  really  means.  A  clock 
goes  more  slowly  when  in  motion  than  when  at  rest; 
the  faster  a  body  moves  the  longer  the  time  intervals. 
As  the  speed  of  a  body  increases,  therefore,  a  clock 
runs  more  and  more  slowly,  and  each  minute  and 
second  becomes  longer  and  longer.  Supposing  that 
as  I  write  these  words,  my  room  could  be  sealed  up 
and  shot  off  into  space  with  a  speed  approaching  that 
of  waves  of  light,  one  hundred  and  fifty,  or  one  hun- 
dred and  seventy  thousand  miles  per  second.  As  the 
speed  of  my  room  increased,  my  desk  clock  would  run 
more  and  more  slowly,  each  tick  would  represent  the 
passing  of  an  hour  or  a  day,  perhaps  even  of  a  year 
of  ordinary  earthly  time.  I  would  not  know  the  dif- 
ference, my  heart  would  beat  regularly,  but  each  beat 
would  mark  the  passing  of  months.  At  the  end  of 
half  an  hour  by  my  clock,  and  before  this  paragraph 
could  be  completed,  my  room  would  have  traversed 
the  depths  of  space  and  been  returned  again  to  my 
island  home.  But,  as  I  glance  up  from  my  paper, 


36     Gravitation  versus  Relativity 

what  a  change  of  scene!  The  peaceful  bay  on  which 
my  windows  give,  would  be  filled  with  strange  craft, 
alien  peoples  would  troop  around,  and  I  would  learn 
that  America  had  decayed  and  fallen,  as  Rome  fell, 
centuries  before.  The  names  of  Harding,  Lloyd 
George,  Poincare  would  be  meaningless;  the  World 
War  even  would  have  been  forgotten,  or  remembered 
only  to  plague  some  schoolboy  with  the  histories  of 
past  and  long  forgotten  races.  This  is  what  relativity 
of  time  really  means. 

Now  motion  takes  place  in  space  and,  if  relativity 
be  true  and  time  varies  with  motion,  time  and  space 
can  no  longer  be  considered  as  independent :  they  are 
bound  together  in  some  way,  neither  can  exist  with- 
out the  other.  A  point  in  space  cannot  be  thought  of 
without  predicating  a  time,  and  an  interval  of  time 
has  no  meaning  except  in  connection  with  a  definite 
moving  body  in  space.  It  is  perfectly  true,  if  such  a 
phrase  be  allowable  in  a  discussion  of  relativity,  that, 
even  under  the  old  concepts  of  time  and  space,  it  is 
extremely  difficult  to  conceive  of  time  without  space, 
or  of  space  without  time;  yet  the  essential  point  of  the 
old  concepts,  the  independence  of  space  and  time,  is 
easy  to  understand :  that  an  instant  of  time  is  the  same 
instant  throughout  all  space,  that  an  interval  of  time 
is  the  same,  whether  measured  in  one  part  of  space 
or  in  another,  upon  a  body  at  rest  or  upon  a  body  in 
motion.  It  is  this  essential  point  in  the  old  theories 


The  Theory  of  Relativity         37 

that  the  relativitist  denies.  Time,  according  to  the 
new  theory,  is  relative,  relative  to  position  in  space 
and  to  the  motions  of  bodies  therein.  An  interval  be- 
tween two  events  is  not  a  fixed,  definite  interval;  it 
is  longer  or  shorter  depending  upon  the  speed  with 
which  the  observer  is  moving  through  space. 

If  space  and  time  are  thus  connected,  it  should  be 
possible  to  express  the  connection  in  terms  of  mathe- 
matical symbols  and  equations.  It  is  at  this  point  and 
for  this  purpose  that  the  formulas  and  methods  of 
Four  Dimensional  Geometry  are  introduced  into  the 
relativity  theory.  To  most  people,  the  very  words, 
four  dimensions,  are  enough;  everything  at  once  be- 
comes incomprehensible  and  absurd.  Yet  there  is  no 
reason  for  this  too  prevalent  idea :  in  the  broad  sense 
of  the  words,  there  is  nothing  new  or  startling  in  the 
four  dimensional  idea.  It  is  a  matter  of  common, 
every-day  knowledge  that,  in  order  to  describe  fully 
an  event,  we  must  tell  not  only  where  the  event  took 
place,  but  when.  To  speak  of  the  Battle  of  the  Marne 
does  not  definitly  fix  the  event,  for  more  than  one 
battle  was  there  fought ;  in  the  World  War  there  were 
at  least  two  distinct  battles  on  the  Marne.  The  when, 
the  date,  is  essential  if  we  are  to  particularize  a  certain 
definite  battle,  or  event.  To  fix  definitely  the  place 
at  which  an  event  occurs  requires  three  elements,  or 
coordinates,  for  all  objects  in  space  have  three  dimen- 
sions, length,  breadth,  and  thickness.  To  locate  the 


38     Gravitation  versus  Relativity 

place  of  the  battle  the  three  place  elements,  or  coordi- 
nates are,  the  surface  of  the  earth,  the  latitude,  and 
the  longitude  of  the  point  on  the  river  at  which  the 
battle  was  fought.  But  to  identify  the  First  Battle, 
or  the  Second  Battle,  or  any  particular  event  of  either 
we  must  have  a  fourth  element,  the  date  on  which  the 
event  happened.  That  is,  to  completely  identify  any 
event,  a  battle  on  the  earth's  surface,  the  fall  of  a 
meteor,  the  collision  of  two  stars,  we  must  have  four 
elements;  three  to  fix  the  position  in  space,  and  one 
to  fix  it  in  time.  Thus,  in  the  broad,  general  sense  of 
the  words,  we  live  and  have  our  being  in  a  world  of 
four  dimensions,  and  mathematically  speaking  it  re- 
quires four  numbers,  statements,  or  coordinates,  to 
identify  fully,  in  space  and  time,  an  event  or  happening. 

There  is  nothing  new  in  all  this.  But  what  is  new, 
what  is  startling,  is  the  point  emphasized  in  the  rela- 
tivity theory,  the  introduction  of  a  definite  mathemati- 
cal relation  between  the  space  coordinates  and  the  timei 
coordinate.  This  relationship  is  introduced  through 
the  adoption  of  the  Lorentz  transformation  equations, 
heretofore  fully  explained,  and  was  specifically  brought 
out  by  Minkowski.  It  can  best  be  understood  by  means 
of  an  illustration. 

In  our  ordinary,  every-day  world  two  surveyors 
compare  their  results  as  to  the  positions  of  two  definite 
points,  two  boundary  stones,  or  markers,  for  example. 
One  surveyor  runs  his  lines  as  his  compass  points, 


The  Theory  of  Relativity          39 

north  and  south,  east  and  west ;  the  other,  knowing  that 
the  compass  does  not  point  towards  the  true  north, 
corrects  his  compass  courses  and  runs  his  lines  parallel 
to  and  at  right  angles  to  the  true  meridian.  The  actual 
measures  of  the  two  surveyors  will  differ,  their  lines, 
as  run,  will  be  of  different  lengths;  but,  if  their  work 
be  accurate,  their  final  results  as  to  the  actual  distance 
between  the  two  points  will  agree.  This  is  shown  in 
the  accompanying  diagram.  The  one  surveyor  meas- 
ures the  lines  AL  and  LB :  the  other  the  lines  AL' 
and  L'B.  Their  systems  of  coordinates  are  different, 
but  the  distance  between  the  two  points,  A  and  B,  is 
the  same. 


Coordinates  and  Distance. 


This  distance,  AB,  may  be  found  from  the  meas- 
urements of  either  surveyor  by  the  aid  of  the  well 
known  problem  of  geometry,  the  problem  which  has 
been  the  bane  of  so  many  students  since  the  time  of 
Euclid,  the  pons  asinorum:  in  a  right  triangle  the 
square  of  the  hypothenuse  is  equal  to  the  sum  of  the 


40     Gravitation  versus  Relativity 

squares  of  the  other  two  sides.  The  first  surveyor 
measures  the  lines  AL  and  LB,  and  from  his  work 
he  concludes: 

(AB)*  =  (AL)'-f  (LB)2 
and  similarly  the  second  surveyor  finds: 

(AB)2  =  (AL')2  +  (L'B)a 

The  distance  between  the  two  points  is  the  one  fixed, 
invariable  quantity:  no  matter  how  many  surveys 
may  be  made,  no  matter  how  the  various  surveyors 
may  run  their  lines,  this  distance,  AB,  comes  out  the 
same,  provided  only  that  the  surveyors'  work  be  accu- 
rate. This  can  be  expressed  mathematically  by  the 
equation, 


where  D  represents  the  distance  between  the  two 
points,  and  x  and  y,  the  measures  by  any  surveyor,  in 
any  two  directions  at  right  angles  to  each  other. 

This  relation,  as  expressed  in  the  above  formula, 
holds  for  any  two  points  situate  on  a  plane  surface, 
upon  the  floor  of  a  room,  upon  small  portions  of  the 
surface  of  the  earth.  And  an  entirely  similar  relation 
holds  for  the  distance  between  two  points  in  space, 
between  the  diagonally  opposite  corners  of  a  room,  or 
between  a  point  on  the  ground  and  the  top  of  a  church 


The  Theory  of  Relativity          41 

spire.  But  in  this  case  three  measures  are  required,  the 
length,  breadth,  and  height  of  the  room  for  example. 
And  if  we  denote  by  z  the  third  distance  measured, 
then  the  distance  between  the  two  points  will  be 
given  by, 


D  =  V 


x'  +  y'-f  z2 

And  this  relation  between  the  measured  coordinates 
and  the  distance  between  the  two  points  holds  true  no 
matter  how  the  three  lines,  or  coordinates,  x,  y,  z, 
are  run,  provided  only  that  they  are  mutually  perpen- 
dicular to  one  another,  like  the  three  corner  lines  of  a 
room.  This  equation,  therefore,  represents  a  definite, 
fundamental  relation  between  the  coordinates  of  points 
in  ordinary  space  :  the  distance  is  the  same,  no  matter 
upon  what  system  the  individual  measures  are  made. 
In  the  terms  of  the  mathematician,  D  is  invariant. 

Now  Minkowski  showed  that,  when  the  Lorentz 
transformation  equations  are  used,  there  is  a  similar 
invariant  quantity  connecting  the  four  coordinates, 
necessary  to  locate  an  event  in  space  and  time.  This 
quantity  is  : 


D'  =     x'+y2-f  z'-c2t2 

where  c  is  the  velocity  of  light  and  t,  the  interval  of 
time  between  the  two  events,  and  x,  y,  z,  the  ordinary 
three  distance  coordinates.  Now  Minkowski  showed 
that,  no  matter  in  what  directions  the  measures  are 


42     Gravitation  versus  Relativity 

made  no  matter  what  system  of  coordinates  be  used, 
then  D'  always  has  the  same  value;  it  is  invariant, 
absolute,  and  thus  furnishes  a  definite  and  fixed  rela- 
tion between  the  space  coordinates  and  the  time  co- 
ordinate. It  has  been  called  the  true  or  "absolute" 
interval  between  two  events : — the  interval  in  time 
and  in  space. 

As  this  so-called  relation  between  the  measure- 
ments in  the  time  and  in  space  is  one  of  the  fundamental 
assumptions  of  the  relativity  theory,  let  us  try  to 
visualize  it  and  see,  if  we  can,  what  it  really  is.  An 
event  happens  in  New  York  City  at  ten  o'clock  in  the 
morning  of  a  certain  day,  a  motor  car  accident  at 
Fifth  Avenue  and  42nd  Street  for  example;  at  four 
o'clock  in  the  afternoon  of  the  same  day  a  second 
accident  takes  place  at  the  same  point  in  the  city. 
These  two  accidents,  happening  at  identically  the  same 
point  in  space  (the  motion  of  the  earth  being  disre- 
garded), are  separated  by  a  time  interval  only,  by  an 
interval  of  six  hours.  Now  a  third  motor  accident 
happens  also  at  four  o'clock,  but  in  Washington  instead 
of  New  York.  This  accident,  or  event,  in  Washing- 
ton is  separated  from  the  second  New  York  accident, 
not  by  a  time  interval,  for  they  both  occur  at  identically 
the  same  instant,  but  by  a  space  interval  or  distance 
of  some  250  miles.  What  is  the  actual,  or  absolute, 
interval  in  time  and  space  between  the  first  accident 
in  New  York  and  the  third  accident  in  Washington? 


The  Theory  of  Relativity          43 

To  this  question  the  ordinary  mortal,  remembering  his 
early  lessons  in  elementary  arithmetic,  would  answer 
that  you  cannot  add  time  to  distance  any  more  than 
you  can  add  horses  to  pigs,  and  that  the  two  accidents 
are  separated  by  a  distance  of  250  miles  and  by  an 
interval  of  six  hours  in  time.  But  the  enlightened 
relativitist  accomplishes  the  seeming  impossibility  by 
the  aid  of  Minkowski's  formula  and  finds  something 
which  he  calls  the  space-time  interval  between  the 
two  events.  He  has  either  converted  the  horses  into 
pigs,  or  the  pigs  into  horses,  or  formed  some  hybrid 
out  of  them  both. 

This  mathematical  expression  of  Minkowski  for  a 
space-time  interval  corresponds  closely  to  our  ordinary 
expression  for  the  distance  between  two  objects,  but 
not  exactly.  The  term  involving  the  time  is  preceded 
by  a  minus  sign  instead  of  a  plus  sign.  The  corre- 
spondence, however,  can  be  made  complete,  if  the  time 
coordinate,  ct,  is  replaced  by  the  imaginary  quantity 
ctX  V—  i.  Then  the  Minkowski  expression  becomes 
identical  with  the  ordinary  distance  formula,  except 
that  it  involves  four  quantities  instead  of  three,  and 
all  the  various  equations  of  the  relativity  theory  assume 
mathematical  forms  in  four  coordinates,  in  which  this 
new  time  coordinate  plays  exactly  the  same  role  as 
the  three  space  coordinates.  But  in  all  these  expres- 
sions and  equations  the  time  coordinate  involves  a 
f actor,  V  -  i.  This  is  the  mathematical  symbol  for 


44     Gravitation  versus  Relativity 

an  imaginary  quantity,  for  something  we  can  neither 
visualize,  nor  conceive  of.  It  is  useless  to  attempt  to 
illustrate  or  visualize  the  connection  between  time  and 
space;  the  very  mathematical  symbol  used  to  denote 
the  form  of  the  connection  indicates  the  impossibility 
of  our  doing  so.  Thus  the  very  mathematical  symbol, 
used  by  the  followers  of  relativity,  indicates  the  purely 
imaginary  character  of  all  their  reasoning. 

From  these  postulates  and  principles  Einstein  has 
built  up  his  entire  theory  of  relativity.  These  postu- 
lates and  principles,  it  must  be  remembered,  are  pure 
assumptions,  assumptions  that  may  appear  to  have 
more  or  less  plausibility,  but  assumptions  nevertheless. 
But  once  these  assumptions  are  accepted  as  true,  it  is 
possible  to  build  upon  them  a  complete  and  logical 
system  of  the  universe  and  of  all  physical  phenomena 
and  happenings  therein.  This  Einstein  has  done  with 
great  technical  skill  and  with  a  broad  view  of  the  many 
intricate  problems  involved.  In  this  development  of 
his  theories  two  stages  are  recognized,  known  respec- 
tively as  the  "special"  and  the  "general"  theories  o£ 
relativity. 

In  the  "special"  theory,  which  antedated  the  "gen- 
eral" by  several  years,  the  motions  of  the  reference 
bodies,  or  sets  of  coordinate  axes,  were  assumed,  or 
restricted,  to  be  uniform,  rectilinear,  and  non-rotary: 
that  is,  all  were  assumed  to  move  forever  in  straight 
lines  at  constant  speed  and  without  rotation  of  any 


The  Theory  of  Relativity          45 

kind.  And  the  basic,  underlying  principle  of  the  theory 
is  that  the  laws  of  mechanics,  of  physics,  all  the  general 
laws  of  nature,  have  exactly  the  same  form  when  re- 
ferred to  every  such  frame  of  reference :  that  for  the 
physical  description  of  natural  phenomena  there  is  no 
"unique"  frame,  no  body  at  absolute  rest.  Involved 
in  this  are  all  the  special  postulates,  or  assumptions 
regarding  the  velocity  of  light,  the  relativity  of  time, 
and  the  space-time  theorem  of  Minkowski,  all  of 
which  have  been  fully  explained  in  the  previous 
pages. 

But  in  the  universe  at  large  there  is  no  body  which 
strictly  complies  with  the  restrictions  of  the  special 
theory;  no  body,  the  motion  of  which  is  uniform,  rec- 
tilinear, and  non-rotary.  All  the  bodies  of  the  solar  sys- 
tem are  moving  in  curved  paths  and  they  are  all  rotat- 
ing. To  not  a  single  body,  therefore,  can  there  be  at- 
tached a  set  of  axes,  or  a  frame  of  reference,  which  fills 
the  required  conditions  of  the  special  theory.  This 
theory  is  thus  an  approximation,  a  particular  case;  a 
case  which  can  never  actually  occur,  but  which  may  be 
approached  so  closely  in  many  physical  problems  of  the 
laboratory,  that  the  errors  introduced  through  its  use 
are  negligible.  But,  when  the  motions  of  the  planets 
themselves  are  considered  in  reference  to  the  universe 
at  large,  then  the  restrictions  of  the  special  theory  must 
be  abandoned. 

This  leads  up  to  the  "general"  theory,  in  which  all 


46     Gravitation  versus  Relativity 

bodies  are  treated  as  being  in  gravitational  fields  and 
as  having  non-uniform,  or  accelerated  motions.  In 
this  broader  theory,  the  basic  principle  of  relativity 
may  be  stated  as:  "All  bodies  of  reference  K,  K' , 
etc.,  are  equivalent  for  the  description  of  natural  phe- 
nomena (formulation  of  the  general  laws  of  nature) , 
whatever  may  be  their  state  of  motion"  (72)  :  or 
again,  "The  general  laws  of  nature  are  expressed 
through  equations,  which  hold  for  all  systems  of  co- 
ordinates" This  broader  theory  involves  all  the  funda- 
mental conceptions  as  to  time  and  space  as  set  forth 
in  the  "special"  or  limited  theory,  but  the  formulas,  or 
equations  expressing  the  results  become  more  general  or 
fluid  in  character. 

Among  all  physical  phenomena  gravitation  stands 
preeminent.  Electric  and  magnetic  phenomena  depend 
upon  the  constituent  material,  and  upon  the  physical 
state  of  a  body,  but  gravitation  is  the  same  for  all 
bodies  and  for  all  conditions  of  the  same  body; 
the  same  for  lead  and  for  a  feather,  the  same  for  ice, 
water,  and  steam.  A  piece  of  lead  and  a  feather  fall 
towards  the  earth  in  exactly  the  same  manner  in  vacuo, 
provided  only  that  they  start  from  the  same  point  in 
space  with  the  same  initial  velocity.  Gravitation 
seems,  therefore,  an  attribute  of  the  particular  point 
in  space,  rather  than  of  the  particular  body  which  hap- 
pens to  occupy  that  point  at  any  instant.  According 
to  Einstein  the  phenomena  can  be  considered  some- 


The  Theory  of  Relativity          47 

what  in  the  following  manner: — a  material  body,  the 
earth  in  the  above  example,  affects  the  space  in  its 
immediate  neighborhood,  gives  it  a  peculiar  twist,  or 
warp,  produces,  in  other  words,  a  gravitational  field. 
The  motions  of  any  body  entering  such  a  field  are 
determined  by  the  laws  which  govern  the  properties 
in  space  of  the  gravitational  field  itself.  What  is  to 
be  considered,  therefore,  is  not  the  motions  of  particu- 
lar bodies,  but  the  characteristics  of  space,  as  affected 
by  the  presence  of  matter. 

Such  study  involves  the  most  intricate  mathematics, 
and  the  mathematical  processes  and  methods,  used  by 
Einstein,  cannot  be  explained  in  untechnical  language. 
The  general  result,  however,  is  that  "the  geometrical 
properties  of  space  are  not  independent,  but  they  are 
determined  by  matter"  (135).  By  geometrical  proper- 
ties is  to  be  understood  the  mathematical  relations  be- 
tween measured  quantities;  between  the  radius  of  a 
circle  and  its  circumference,  between  the  area  of  a 
square  and  the  length  of  its  sides.  Since  the  time  of 
Euclid  we  have  been  taught  to  think  that  for  every 
circle,  wheresoever  situated,  on  the  earth,  about  the 
sun,  near  Venus,  or  in  the  vicinity  of  the  North  Star, 
the  circumference  is  3.141592+  times  the  radius.  Not 
so  in  the  relativity  theory,  every  gravitational  field  has 
its  own  system  of  geometry.  Near  Venus  this  ratio 
may  have  one  value,  at  the  North  Star  an  entirely  dif- 
ferent value.  On  the  earth  the  area  of  a  square,  the 


48     Gravitation  versus  Relativity 

sides  of  which  are  two  feet  long,  is  four  square  feet. 
According  to  relativity  the  area  of  such  a  square,  if 
transported  to  the  sun,  would  not  be  four  square  feet, 
but  something  entirely  different:  in  Orion  this  area 
would  have  a  still  different  value.  Thus,  if  the  relativ- 
ity theory  be  true,  the  formulas  and  methods  of  geome- 
try and  of  engineering  to  which  we  are  accustomed  hold 
only  for  the  Earth;  the  inhabitants  of  Mars,  if  any 
there  be,  have  a  different  geometry  and  different  formu- 
las to  solve  their  engineering  problems. 

Under  the  relativity  theory  the  mathematical  expres- 
sion for  the  law  of  gravitation  is  not  the  same  as  that 
formulated  by  Sir  Isaac  Newton.  This  is  necessarily 
so,  for,  in  accordance  with  the  principles  of  relativity, 
the  velocity  of  a  body  enters  into  every  formula  and 
into  every  measure  of  its  position  in  space.  This  was 
seen  in  the  Lorentz  transformation  equations,  which 
contain  terms  involving  the  ratio  of  the  velocity  of  the 
body  to  that  of  light.  The  law  of  Einstein  differs, 
therefore,  from  that  of  Newton  by  the  introduction  of 
terms  depending  upon  this  ratio  of  velocities.  When 
the  velocity  of  a  body  is  extremely  small  as  compared 
to  the  velocity  of  light,  then  this  ratio  becomes  small 
and  these  terms  become  negligible  in  comparison  with 
the  other  terms  of  the  expression.  In  this  case  the 
formulas  of  Einstein  degenerate  into  those  of  Newton, 
and,  thus,  for  small  velocities  the  two  laws  give  identi- 
cally the  same  results. 


The  Theory  of  Relativity         49 

Besides  this  change  in  the  law  of  gravitation,  there 
are  many  logical  deductions  from  the  postulates  and 
principles  of  relativity.  Among  the  principal  ones  may 
be  mentioned : 

1.  The  gravitational  mass  of  a  body  is  equal  to 
its  inertia!  mass. 

2.  In   gravitational    fields    there    are    no    such 
things  as  rigid  bodies  with  Euclidean  proper- 
ties. 

3.  Light  is  subject  to  gravitation,  and  in  gravi- 
tational fields  rays  of  light  travel  in  curved 
paths. 

4.  The  lines  of  the  solar  and  stellar  spectra  are 
displaced  towards  the  red  end  of  the  spec- 
trum as  compared  with  the  spectral  lines  of 
the  same  element  produced  on  the  surface  of 
the  earth. 

The  following  table  exhibits  a  few  of  the  principal 
differences  between  the  so-called  classical,  or  standard 
theories  and  the  relativity  theory,  as  enunciated  by 

Einstein : 

Standard  Theories      Einstein   Theory 

Space:  Independent  &  ab-    Dependent:      con- 

solute,  nected  to  and  in- 

volved with  time. 

Time:  Independent  &  ab-  Dependent  upon 
solute :  same  every-  space :  varies  with 
where  and  under  positions  and  mo- 
all  conditions.  tions  of  bodies. 


5°      Gravitation  versus  Relativity 


Time  intervals :  Identical  every-  Vary  with  the  posi- 
where  and  under  tions  and  motions 
all  conditions.  of  bodies. 


Rigid  bodies : 


Geometry : 


Of     same     dimen-     Vary    in    size    and 
sions     and     shape     shape  with  motion, 
under     all     condi 
tions   of    motion. 


Laws      and 
mulas      the 
everywhere 
under     all 
tions. 


for- 

same 

and 

condi- 


Speed  of  light:   Constant  in  space. 


Ray  of  light : 


Gravitation : 


Travels  in  straight 
lines. 


Independent         of 
motion  of  bodies. 
Due  to  the  attrac- 
tion    of     material 
bodies. 


Laws  and  formulas 
vary  under  gravita- 
tional action  of  ma- 
terial bodies. 


Appears  the  same 
to  every  observer, 
whatever  his  mo- 
tion. 

Travels  in  curved 
paths  under  attrac- 
tion of  material 
bodies. 

Varies  with  the 
speed  of  bodies. 
Material  bodies 
warp  space,  and 
this  "warp"  causes 
motion  in  bodies. 


The  fundamental  postulates,  or  assumptions  of  rela- 
tivity are  so  broad  and  general  that  it  is  next  to  im- 
possible to  directly  test  their  truth  or  falsity.  But, 


The  Theory  of  Relativity          51 

indirectly  the  theory  can  be  tested  through  conclusions, 
or  formulas  which  have  been  logically  derived  from  its 
basic  principles.  Einstein,  himself,  has  devised  certain 
physical  and  astronomical  tests  of  his  theories,  and  has 
claimed  that  such  tests  have  been  successfully  met  and 
conclusively  prove  the  truth  of  the  entire  theory.  Such 
claims  and  such  tests  are  fully  explained  in  the  follow- 
ing chapter. 


CHAPTER  II 

THE    EVIDENCE    FOR    THE    RELATIVITY    THEORY 

THE  RELATIVITY  THEORY  strikes  directly  at  our  fun- 
damental concepts  as  to  the  structure  of  the  universe ; 
its  conclusions  are  startling  and  completely  upsetting 
to  our  ordinary  common-sense  way  of  looking  at  phys- 
ical and  astronomical  phenomena.  To  have  such  a 
theory  accepted,  it  would  seem  that  the  evidence  in  its 
favor  must  be  overwhelming,  that  the  experiments, 
cited  by  its  supporters,  must  be  clear-cut  and  admit  of 
no  other  solution.  The  burden  of  proof  should  be  on 
the  relativitist,  and  it  should  be  clearly  shown  in  each 
case  or  experiment,  cited  by  him,  that  the  relativity 
theory  is  the  necessary  and  sufficient  explanation;  it 
should  be  established  beyond  all  reasonable  doubt,  not 
only  that  the  phenomena  can  be  explained  by  the  rela- 
tivity theory,  but  that  no  other  hypothesis  or  theory 
can  equally  well  account  for  the  observed  facts. 

Has  this  been  done?  Do  the  experiments  and  phe- 
nomena, cited  by  Einstein  clearly  establish  the  truth 
of  his  theories  by  excluding,  as  possible  explanations, 
all  other  hypotheses  and  theories?  In  addition  to  the 

5* 


Evidence  for  Relativity  Theory   53 

"decisive"  experiment  of  Michelson  and  Morley,  Ein- 
stein claims  four  experiments,  or  observations,  as  fully 
and  completely  confirming  his  theories.  Two  of  these 
might  be  classed  as  physical  experiments,  two  as 
astronomical  observations.  These  four  are : 

1.  The  Fizeau  experiment  on  the  velocity  of 

light  in  a  stream  of  flowing  water. 

2.  The  shift  in  the  lines  of  the  solar  spectrum. 

3.  The  motion  of  the  perihelion  of  Mercury. 

4.  The  deflection  of  light  waves,  as  observed 

in  the  eclipse  of  1919. 

The  first  two  of  these  so-called  proofs  are  physical 
and  belong  primarily  to  the  realm  of  experimental 
physics,  to  the  realm  of  the  laboratory;  the  latter  two, 
on  the  contrary,  are  purely  astronomical  and  must  be 
studied  and  judged  by  astronomical  methods.  The 
astronomical  observations  are  the  ones  relied  upon, 
mainly,  as  furnishing  the  proof  of  the  Einstein  theories ; 
that  of  the  perihelion  of  Mercury  being  the  first  an- 
nounced by  Einstein  and  the  one  most  widely  quoted. 
The  observations  of  the  British  astronomers  at  the 
1919  eclipse,  however,  are  equally  important,  are  more 
striking,  and  more  easily  understood.  It  was  the  an- 
nouncement of  the  results  of  these  observations  that 
caused  the  wide-spread,  popular  interest  in  Einstein 
and  the  relativity  theory.  These  astronomical  observa- 
tions are  fully  explained  and  discussed  in  the  following 
pages. 


54     Gravitation  versus  Relativity 

The  physical  experiments,  or  observations,  on  the 
other  hand,  have  so  far  yielded  little  evidence  for  or 
against  the  theories  of  Einstein,  and  are,  therefore, 
but  very  briefly  treated.  Sufficient  outlines  of  the 
experiments,  however,  are  given  to  enable  the  reader 
to  form  a  judgment  as  to  the  character  of  the  evidence 
and  of  the  methods  of  reasoning  adopted  by  the 
relativitist. 

i.     THE  FIZEAU  EXPERIMENT 

The  Fizeau  experiment  was  first  made  in  1859. 
It  has  since  been  repeated  in  its  original  form  and 
with  modified  and  improved  apparatus,  with  results 
always  substantially  the  same  as  those  obtained  by 
Fizeau  some  twenty  years  before  Einstein  was  born. 

This  experiment  of  Fizeau  was  a  straight- forward, 
clean-cut  attempt  to  verify  certain  predictions  made  by 
Fresnel  as  to  the  speed  of  light  in  different  transparent 
substances,  when  the  substances,  or  media  are  at  rest 
and  when  in  motion.  These  predictions  had  been 
embodied  in  what  has  come  to  be  known  as  Fresnel's 
law :  that  the  velocity  of  light  in  a  moving  transparent 
medium  depends  upon  the  speed  with  which  the  medium 
is  moving  and  upon  its  index  of  refraction.  This 
index  of  refraction,  it  will  be  remembered,  is  a  definite 
optical  property,  or  characteristic  of  a  transparent 
medium;  being,  in  fact,  the  ratio  of  the  speed  of  a 
ray  of  light  in  vacuo  to  the  speed  of  the  ray  in  the 


Evidence  for  Relativity  Theory    55 

medium.  This  index  varies  for  different  substances, 
being  different  for  air,  for  water,  for  glass,  different 
for  the  various  varieties  of  glass.  Now  in  testing 
this  law  of  Fresnel,  Fizeau  sent  a  beam  of  light 
through  a  stream  of  water,  which  was  flowing  through 
a  tube  with  a  definite  and  controlled  velocity.  He 
found  that  the  velocity  of  such  a  beam  of  light  is 
increased  or  decreased  according  as  it  travels  with  or 
against  the  stream,  and  he  found  that  this  increase 
or  decrease,  within  the  .limits  of  accurate  measure- 
ment, always  bears  the  definite  relation  to  the  index 
of  refraction  and  to  the  velocity  of  the  water  required 
by  Fresnel's  law. 

Thus  the  results  of  Fizeau's  experiments  are  in 
accord  with  the  heretofore  accepted  laws  of  optics. 
Sir  Oliver  Lodge  finds  them  to  be  in  strict  accord  with 
classical  ideas  of  a  stationary  ether  and  with  the  ordi- 
nary laws  of  optical  phenomena.  Lorentz  also  gave 
a  satisfactory  theoretical  explanation  of  these  results, 
basing  his  work  on  the  newer  ideas  of  the  electro- 
magnetic structure  of  matter. 

The  relativity  theory  can,  however,  also  explain  the 
results  of  these  experiments.  By  using  approxima- 
tions and  discarding  certain  small  terms  as  negligible, 
Einstein  succeeds  in  bringing  his  formulas  into  close 
accord  with  the  observed  facts,  and  in  showing  that 
these  experiments  do  not  invalidate  his  theories  (48). 
But  the  fact  that  Lorentz  had  fully  explained  the 


56     Gravitation  versus  Relativity 

phenomena  long  before  the  relativity  theory  was  formu- 
lated "does  not  in  the  least,"  according  to  Einstein, 
"diminish  the  collusiveness  of  the  experiment  as  a 
crucial  test  in  favor  of  the  theory  of  relativity,  for 
the  electrodynamics  of  Maxwell-Lorentz,  on  which 
the  original  theory  zvas  based,  in  no  way  opposes  the 
theory  of  relativity.  Rather  has  the  latter  been  de- 
veloped from  electrodynamics  as  an  astoundingly 
simple  combination  and  generalization  of  the  hypothe- 
ses, formerly  independent  of  each  other,  on  which 
electrodynamics  was  built"  (48). 

These  two  sentences  of  Einstein  are,  from  one  point 
of  view,  as  important  as  any  in  his  work  on  relativity : 
— they  should  be  read  and  re-read.  They  give  a  direct 
insight  into  his  methods  of  reasoning.  Here  is  an 
experiment,  the  details  do  not  matter,  an  experiment 
claimed  by  Einstein  as  a  "crucial  test"  of  his  theories, 
yet  in  the  very  sentence,  in  which  this  claim  is  advanced, 
he  admits  that  other  theories,  the  very  theories  he 
attempts  to  overthrow,  can  equally  well  explain  the 
phenomenon.  How  can  an  experiment,  equally  well 
explained  by  several  different  theories,  be  a  "crucial 
test"  in  favor  of  one  of  them? 

2.     SHIFT  OF  SPECTRAL  LINES 

According  to  deductions  and  calculations  based 
upon  the  relativity  theory,  all  the  lines  of  the  solar 
spectrum  should  be  displaced  slightly  towards  the  red 


Evidence  for  Relativity  Theory     57 

end  of  the  spectrum,  when  compared  with  similar  lines 
obtained  from  terrestrial  sources.  It  is  well  known 
that  the  light  emitted  by  any  element  when  rendered 
incandescent,  consists  of  a  series  of  waves  of  different 
lengths,  and  that,  when  such  light  is  analyzed  by,  or 
spread  out  in,  a  spectroscope,  these  different  sets  of 
waves  appear  as  separate  bands  or  lines  of  light.  Each 
element  has  its  own  characteristic  series  of  lines  by 
which  it  can  be  recognized,  whether  it  be  in  the  flame 
of  a  candle,  in  an  electric  arc,  in  the  atmosphere  of 
the  sun,  or  that  of  a  distant  star.  Under  certain  con- 
ditions these  lines  appear  as  bright  streaks  of  color 
against  a  dark  back-ground;  under  other  conditions, 
as  dark  lines  crossing  the  brilliantly  colored  spectrum. 
But,  light  or  dark,  the  position  of  a  particular  line  is 
the  same,  and  it  is  the  relative  position  of  the  line  in 
the  spectrum  which  identifies  it. 

The  position  of  a  line  in  the  spectrum  is  not  always 
absolutely  the  same:  it  shifts  slightly  towards  the  blue 
or  towards  the  red  end  as  the  source  of  light  is 
approaching  towards  or  receding  from  the  observer 
This  is  illustrated  in  the  accompanying  figure,  which 
shows  in  the  upper  portion  the  brilliant  sodium  lines 
as  produced  in  the  ordinary  spectrum  on  the  earth,  and 
in  the  lower  portion  the  reversed  spectrum  of  an 
imaginary  star. 

If  the  star  and  the  earth  were  at  rest  relative  to 
each  other,  then  the  dark  lines  of  the  stellar  spectrum 


58     Gravitation  versus  Relativity 

would  form  exact  continuations  of  the  bright  terrestrial 
lines.     As,  however,  the  stellar  lines  appear  shifted 


Fig.  6.    Displacement  of  Spectral  Lines. 

towards  the  blue  end  of  the  spectrum,  the  star  is  moving 
towards  the  earth.  The  amount  of  the  shift  depends 
upon  and  is  a  measure  of  the  speed  of  the  approach 
of  the  star.  Thus  by  comparing  the  lines  in  the 
spectrum  of  the  sun,  or  of  a  star,  with  similar  lines 
produced  by  an  electric  arc  in  the  laboratory,  the 
speed  with  which  the  body  is  approaching,  or  receding 
from,  the  earth  can  be  measured.  This  fact,  or  prin- 
ciple, has  been  known  for  many  years,  and  has  been 
in  constant  use  in  determining  the  motions,  to  and 
fro,  of  the  heavenly  bodies. 

Now  Einstein  has  shown  that,  according  to  rela- 
tivity, there  should  be  an  additional  shift  of  the  lines 
in  the  solar  spectrum  towards  the  red  end  of  the 
spectrum.  This  Einstein  shift  is  extremely  small;  the 
shift  for  each  line  being  proportional  to  its  wave 
length  and  amounting  to  about  the  two  millionth  part 
thereof.  Thus,  in  other  words,  if  the  sun  be  approach- 
ing the  earth,  then  the  Einstein  shift  and  the  motion 
shift  act  in  opposite  directions  and  the  total  shift  will 


Evidence  for  Relativity  Theory    59 

be  slightly  less  than  that  due  to  the  motion  alone; 
while,  if  the  sun  be  moving  away  from  the  earth,  then 
the  Einstein  shift  would  be  added  to  the  motion  shift, 
and  the  total  would  be  slightly  greater  than  that  due 
to  the  motion  alone.  Thus  the  Einstein  shift,  if  there 
be  such  a  thing,  is  all  tangled  up  with  the  shift  due 
to  the  relative  motion  of  the  sun  and  earth;  it  is 
extremely  small,  is  at  the  very  limit  of  measurement, 
and  can  only  be  detected  with  the  most  powerful 
modern  instruments  and  with  the  utmost  refinements 
in  method  and  care  in  making  the  observations.  So 
difficult  are  the  measurements  and  so  surrounded  by 
disturbing  factors,  that  it  seems  hardly  possible  that 
a  conclusive  result  can  be  obtained. 

Some  indications  of  an  Einstein  shift  have  been 
found  by  certain  observers,  but  the  shift  so  found, 
if  real,  does  not  agree  in  amount  with  that  required 
by  the  relativity  theory.  Various  lines  in  the  spectrum 
give  radically  different  results,  and  different  observers 
find  different  results  for  the  same  line.  While  ac- 
knowledging these  facts,  Einstein  announces  his  con- 
clusions in  the  following  words:  'Whereas  Grebe 
and  Bachem  (Bonn),  as  a  result  of  their  own  measure- 
ment and  those  of  Evershed  and  Schwarzschild  on  the 
cyanogen  bands  have  placed  the  existence  of  the  effect 
almost  beyond  doubt,  other  investigators,  particularly 
St.  John,  have  been  led  to  the  opposite  opinion  in  con- 
sequence of  their  measurements"  (158). 


60     Gravitation  versus  Relativity 

Here  again  is  a  typical  example  of  the  methods  and 
reasoning  of  the  relativitists.  For,  in  considering  this 
statement  of  Einstein,  it  should  be  remembered  that 
St.  John  made  his  observations  at  the  Mount  Wilson 
Solar  Observatory,  with  an  equipment  far  surpassing 
anything  to  be  found  elsewhere,  while  the  observations 
at  Bonn  were  made  with  the  ordinary,  average  equip- 
ment of  a  small  laboratory  or  observatory. 

3.     THE  MOTION  OF  MERCURY 

In  the  motion  of  Mercury  Einstein  finds,  how- 
ever, a  complete  confirmation  of  the  theory  of  rela- 
tivity, a  confirmation  of  the  necessity  as  well  as  of  its 
sufficiency. 

It  is  well  known  and  has  been  known  for  many 
years,  that,  in  its  motion  about  the  sun,  the  planet 
Mercury  exhibits  a  certain  small  irregularity,  or  dis- 
cordance, the  exact  cause  of  which  has  troubled  two 
generations  of  astronomers.  The  cause  of  this  dis- 
cordant motion,  and  a  full  explanation  of  it,  is  found 
by  Einstein  in  the  formulas  of  the  relativity  theory, 
and  thus  the  relativity  theory  "has  already  explained 
a  result  of  observation  in  astronomy,  against  which 
classical  mechanics  is  powerless"  (121). 

Disregarding  the  action  of  other  bodies  of  the  uni- 
verse, a  planet,  under  the  Newtonian  formulas  of 
classical  mechanics,  travels  about  the  sun  in  an  elliptical 
orbit,  the  major  axis  of  which  remains  permanently 


Evidence  for  Relativity  Theory    61 

fixed  in  direction.  Now  in  1859  Leverrier  found  from 
observations  of  Mercury  that,  after  due  allowance  for 
the  action  of  the  other  known  bodies  of  the  solar 
system,  the  orbit  of  this  planet  does  not  remain  sta- 
tionary, but  slowly  rotates  about  the  sun,  as  shown 
in  the  accompanying  figure. 


Fig.  7.    The  Rotation  of  Mercury's  Orbit. 

In  this  diagram  the  line  PA  is  the  axis  of  the 

elliptic  path  in  which  the  planet  travels  about  the  sun,  S. 

The  point  of  closest  approach  to  the  sun,  P,  is  called 

the  "perihelion,"  and  it  is  this  point  which  it  is  cus- 


62      Gravitation  versus  Relativity 

ternary  to  use  in  fixing  the  direction  of  the  axis  PA. 
Now  Leverrier  found  that,  in  the  case  of  Mercury, 
the  whole  orbit  is  swinging  around  the  sun,  as  on  a 
pivot,  so  that  after  a  lapse  of  ages  the  orbit  will  be 
found  to  lie  in  the  position  of  the  dotted  ellipse,  and 
the  point  P  to  have  moved  forward  to  the  point  P'. 
This  motion  is  extremely  small,  the  point  P  moving 
forward,  according  to  Leverrier,  only  a  few  seconds 
of  arc  in  one  hundred  years.  It  would  thus  require 
some  thirty-three  thousand  centuries  for  the  orbit  to 
make  one  complete  rotation. 

At  the  time  of  Leverrier  and  for  many  years  there- 
after, this  rotation,  which  has  since  been  confirmed  by 
Newcomb,  was  thought  to  be  due  to  the  action  of  a 
planet  between  Mercury  and  the  sun,  and  the  pro- 
visional name  of  Vulcan  was  given  to  such  hypotheti- 
cal planet.  Long  continued  search,  however,  failed 
to  locate  Vulcan,  and  it  is  now  recognized  by  all 
astronomers  that  Vulcan  does  not,  and  never  did,  exist. 
Other  explanations  of  the  observed  rotation  have  been 
offered  from  time  to  time;  but,  according  to  Einstein, 
"This  effect  can  be  explained  by  means  of  classical 
mechanics  only  on  the  assumption  of  hypotheses  which 
have  little  probability,  and  which  were  devised  solely 
for  the  purpose"  (123). 

The  contrary  is  the  case  with  the  relativity  theory. 
Direct  deductions  from  the  Einstein  law  of  gravita- 
tion, without  the  introduction  of  any  new  or  special 


Evidence  for  Relativity  Theory    63 

factors,  show  that  the  orbits  of  all  planets  should 
rotate  in  the  manner  found  for  Mercury,  and  further 
show  that,  in  the  case  of  Mercury,  this  rotation  should 
be  at  the  rate  of  43  seconds  of  arc  per  century. 

This  figure  agrees  almost  exactly  with  that  found 
by  Newcomb  in  his  first  revision  of  Leverrier's  work. 

While  theoretically,  under  the  principles  of  rela- 
tivity, all  the  other  planets  of  the  solar  system  should 
show  similar  orbital  rotations,  the  actual  amounts  of 
such  rotations  decrease  very  rapidly  with  the  increased 
distances  of  the  planets  from  the  sun.  In  all  the  other 
planets  the  magnitude  of  this  motion  is  so  small  as  to 
"necessarily  escape  detection,"  except  possibly  in  the 
case  of  Venus.  The  orbit  of  this  planet,  however, 
is  almost  an  exact  circle,  which  makes  it  more  difficult 
to  locate  the  perihelion  with  precision  (152).  Thus  the 
orbits  of  all  the  planets,  except  that  of  Mercury,  should 
remain  apparently  fixed  in  direction,  should  show  no 
measurable  rotation,  or  departure  from  the  motions 
predicted  by  the  Newtonian  law.  This  according  to 
Einstein  "has  been  confirmed  for  all  the  planets  save 
one,  with  the  precision  that  is  capable  of  being  obtained 
by  the  delicacy  of  observation  attainable  at  the  present 
tune.  The  sole  exception  is  Mercury,  the  planet  which 
lies  nearest  the  sun"  (122). 

Thus  the  motions  of  the  planets,  according  to  Ein- 
stein, appear  to  confirm  with  remarkable  precision  de- 
ductions from  the  relativity  theory  and  to  show  that 


64     Gravitation  versus  Relativity 

this  theory  approximates  the  truth  far  more  closely 
than  the  Newtonian  law  of  gravitation. 

4.     CURVATURE  OF  LIGHT  RAYS 

That  rays  of  light  from  distant  stars  are  bent 
and  deflected  into  curved  paths,  when  passing  near  the 
sun,  appears  to  have  been  proved  by  certain  photo- 
graphs taken  in  connection  with  the  solar  eclipse  of 
May  29,  1919.  It  has  been  seen  that  Einstein's  theories 
call  for  such  a  deflection.  He  predicted,  in  fact,  many 
months  before  the  eclipse  took  place  that  the  deflection 
would  amount  to  1.75  seconds  of  arc  for  light  rays 
just  grazing  the  sun's  surface. 


Fig.  8.    Deflection  of  Light  Rays  by  the  Sun. 

This  is  shown  in  the  above  diagram,  where  M  is 
the  sun  and  S  and  E,  the  star  and  earth  respectively. 
The  ray  from  the  star  starts  out  in  the  direction 
SME'  and,  if  the  sun  were  not  present,  the  star  would 
be  seen  from  the  earth  in  the  reverse  direction,  E'MS. 
The  moment  the  ray  enters  the  gravitational  field  of 
the  sun,  however,  it  is,  according  to  Einstein,  deflected 
into  a  curved  path,  being  bent  apparently  around  the 
sun.  The  light  finally  reaches  the  observer  on  the 


Evidence  for  Relativity  Theory    65 

earth  at  E,  and  to  him  the  star  appears  to  be  at  S'. 
The  deflection  is  the  angle  EME',  and,  as  the  intensity 
of  the  gravitational  field  decreases  with  increased  dis- 
tance from  the  centre  of  the  sun,  so  the  amount  of 
this  deflection  will  also  decrease  with  the  apparent 
distance  of  the  star  from  the  sun. 

If  in  the  course  of  its  annual  path  through  the 
heavens,  the  sun  should  chance  to  come  approximately 
between  the  apparent  positions  of  two  stars,  then 
would  the  rays  from  each  star  be  deflected  and  the 
stars  would  appear  at  a  greater  distance  apart  than 
normally.  On  an  ordinary  night,  when  the  sun  is  in 
another  part  of  the  heavens,  two  stars  might  appear 
as  at  S  and  S',  in  the  accompanying  figure.  When, 
however,  the  sun  reaches  the  neighborhood  of  the 
stars  at  M,  the  rays  from  each  will  be  deflected  radially 
from  the  sun's  centre,  and  the  two  will  appear  to  be 
at  D  and  D'  respectively. 

............. 


Fig.  9.    True  and  Deflected  Positions  of  Stars. 

Now  the  sun  is  so  brilliant  that  ordinarily  stars  in 
its  immediate  vicinity  can  neither  be  seen  nor  photo- 
graphed. For  the  few  moments  of  a  total  eclipse, 


66     Gravitation  versus  Relativity 

however,  the  moon  cuts  off  all  the  direct  sun-light, 
and,  during  this  short  interval,  the  stars  may  be  photo- 
graphed, and  their  respective  positions  determined  from 
such  photographs.  These  positions  can  be  compared 
with  standard  positions,  determined  from  similar 
photographs  made  when  the  sun  was  in  another  part 
of  the  sky. 

In  order  to  test  in  this  way  the  prediction  of  Ein- 
stein and,  through  it,  the  relativity  theory  itself,  The 
Royal  Society  and  The  Royal  Astronomical  Society 
equipped  expeditions  and  sent  them  to  Brazil  and  to 
the  island  of  Principe,  near  the  west  coast  of  Africa. 
These  expeditions  made  numerous  photographs  of  the 
stars  in  the  immediate  vicinity  of  the  sun  during  the 
eclipse  of  May  29,  1919.  Similar  plates  were  taken 
of  the  same  stars,  with  the  same  apparatus,  at  other 
periods  of  the  year  when  the  sun  was  in  another  part 
of  the  heavens.  The  relative  positions  of  the  stars, 
as  determined  from  the  eclipse  plates,  were  compared 
with  the  positions  as  determined  from  the  second  set 
of  what  might  be  called  standard  plates. 

Such  comparison  showed  clearly  that  the  apparent 
positions  of  the  stars,  as  shown  on  the  plates,  were 
different  on  the  day  of  the  eclipse  from  what  they 
were  when  the  sun  was  not  in  that  portion  of  the  sky. 
The  expeditions  appear  to  have  proved  the  existence 
of  light  deflections  caused  by  the  presence  of  the  sun. 
These  observed  deflections,  however,  did  not  agree 


Evidence  for  Relativity  Theory    67 

exactly  with  the  prediction  of  Einstein,  who  had  placed 
the  expected  deflection  at  1.75  seconds  of  arc. 

The  report  containing  the  results  of  the  expeditions, 
as  submitted  to  the  Societies  by  the  Astronomer  Royal 
of  England  and  the  astronomers  in  charge,  Eddington, 
Crommelin,  and  Davidson,  shows  that  the  mean  result 
of  the  plates  taken  at  Brazil  gave  a  deflection  of  1.98 
seconds,  that  the  plates  taken  at  the  island  of  Principe, 
under  unfavorable  conditions  of  cloud,  gave  1.61 
seconds.  The  conclusion  of  this  committee  of  most 
able  and  noted  astronomers  is  given  in  the  report  in  the 
following  words: 

"Both  these  results  point  to  the  full  deflection 
l"-75  °f  Einstein's  generalized  relativity  theory, 
the  Sobral  results  definitely,  and  the  Principe  re- 
sults perhaps  with  some  uncertainty."  * 

According  to  Einstein  himself,  "The  results  of  the 
measurements  confirmed  the  theory  in  a  thoroughly 
satisfactory  manner"  (115). 

Thus  two  astronomical  phenomena,  radically  dif- 
ferent in  character,  appear  to  confirm  the  Einstein 
theories  and  deductions  in  a  most  brilliant  and  satis- 
factory manner.  It  would  seem,  therefore,  as  a  direct 
result  of  astronomical  research,  that  we  must  accept 

*  "A  Determination  of  the  Deflection  of  Light  by  the  Sun's 
Gravitational  Field,"  by  F.  W.  Dyson,  A.  S.  Eddington,  and  C. 
Davidson.  Memoirs,  R.A.S.  62. 


68     Gravitation  versus  Relativity 

the  relativity  theory,  with  all  its  implications  as  to 
warped  space  and  variable  time.  On  the  other  hand, 
the  two  physical  phenomena,  cited  by  Einstein,  fur- 
nish practically  no  evidence  in  favor  of  the  theory. 
Further  refinements  in  method  and  advances  in  instru- 
mental design  may,  however,  enable  physicists  to  deter- 
mine the  presence  or  absence  of  the  displacement  of 
the  spectral  lines,  and  thus  either  reinforce  and  con- 
firm the  astronomical  evidence,  or  place  the  entire 
theory  into  the  category  of  a  brilliant,  though  un- 
confirmed hypothesis. 

At  the  present  moment  the  only  tangible  evidence 
in  favor  of  the  theory  is  that  furnished  by  the  motion 
of  Mercury  and  the  observed  deflection  of  light.  This 
evidence,  certainly  as  stated  by  Einstein,  makes  a 
strong  prima  facie  case  for  the  theory  of  relativity; 
but,  before  the  theory  be  accepted  as  proved,  this 
evidence  should  be  carefully  examined  in  all  its  details. 
Many  a  well  built-up  case  has  completely  collapsed 
when  the  evidence  is  sifted  and  the  witnesses  examined. 

In  the  following  pages  the  evidence  in  the  case, 
Gravitation  versus  Relativity,  as  hereinbefore  presented 
by  the  witnesses  for  Relativity,  is  subjected  to  a  search- 
ing examination. 


CHAPTER  III 

THE   LAW    OF    GRAVITATION 

THE  LAW  OF  GRAVITATION,  as  announced  in  1686  by 
Isaac  Newton,  is  simply  that  every  particle  of  matter 
in  the  universe  attracts  every  other  particle  with  a 
force  proportional  to  the  product  of  their  respective 
masses  and  diminishing  as  the  square  of  their  distance 
apart  increases.  This  law  of  attraction  was  deduced 
from  experience,  from  experiment,  and  does  not 
attempt,  in  any  way,  to  explain  how  the  force  acts 
or  why  it  acts;  it  merely  states  that  such  a  force  does 
act,  and,  in  acting,  follows  a  certain  definite  mathe- 
matical law. 

This  brief  and  simple  statement  of  Isaac  Newton 
summarized  the  experience  of  mankind  from  the 
earliest  ages;  codified  and  united  the  researches  and 
experiments  of  philosophers  and  scientists  from  the 
days  of  Hipparchus.  In  the  methods  and  discoveries 
of  this  most  illustrious  predecessor  of  Newton  are  to 
be  found  the  basic  principles  of  astronomical  research 
and  mathematical  methods.  Living  in  the  second  cen- 
tury before  the  Christian  Era,  he  discarded  the  vague 

69 


70     Gravitation  versus  Relativity 

speculations  as  to  the  primal  cause  of  the  movements 
of  the  sun,  moon,  and  planets  and  substituted  a  study 
of  the  movements  themselves;  devised  mathematical 
formulas  and  theories  to  represent  these  movements, 
to  keep  track  of  the  positions  of  the  bodies  in  the  past, 
and  to  predict  their  places  in  the  future.  He  invented 
the  science  of  trigonometry,  he  elaborated  the  idea  of 
the  epicycle  and  used  this  powerful  mathematical 
method  for  representing  the  motions  of  the  sun  and 
moon.  Although  the  names,  epicycle  and  epicyclic 
theory,  have  long  fallen  into  disrepute,  yet  the  fact  re- 
mains that  the  mathematical  method  devised  by  Hip- 
parchus  under  this  name  is  still  in  use  among 
astronomers  and  physicists,  and  forms  the  basis  of  the 
most  modern  tables  of  the  motions  of  the  sun,  moon, 
and  planets.  The  name  has  been  changed,  that  is  all. 

As  the  epicycle  has  played,  and  still  plays,  such  a 
prominent  part  in  the  theories  of  celestial  motions,  it 
should  be  thoroughly  understood.  It  is  merely  a 
mathematical  device  for  analyzing  and  keeping  track 
of  irregular  motions,  as  is  shown  in  the  accompany- 
ing figure.  Suppose  a  body,  the  sun  for  example,  to 
travel  at  a  uniform  rate  of  speed  in  the  circumference 
of  the  small  circle,  the  centre  of  which  is  c.  At  the 
same  time,  this  centre,  c,  is  moving  forward  at  a 
constant  rate  of  speed  in  the  large  circle  about  E. 
In  the  first  position,  as  shown  in  the  diagram,  the 
sun  will  appear  from  E  to  be  in  the  direction  of  c; 


The  Law  of  Gravitation          71 

but  at  a  later  time  when  S  has  completed  one-half 
revolution,  and  c,  one-quarter  revolution  in  their 
respective  circles,  the  sun  will  have  reached  the  point, 


Fig.  10.    Epicyclic  Motion. 

S',  and  from  the  earth  it  will  appear  in  the  direction 
ES',  in  advance  of  c',  as  indicated  in  position  2.  Thus 
during  this  quarter  revolution  the  sun  will  apparently 
have  been  travelling  faster  than  c.  During  the  next 
quarter  revolution  of  c,  the  sun  will  travel  in  the 
small  circle  to  S",  and  again  the  sun  and  c  will  appear 
in  the  same  line,  as  viewed  from  E.  In  this  quarter 
revolution  the  sun  has  apparently  travelled  more  slowly 
than  has  c.  Upon  the  completion  of  one  entire  revolu- 
tion of  c,  the  sun  and  c  will  return  to  their  positions 


72     Gravitation  versus  Relativity 

which  they  held  at  the  beginning  and  will  again  appear 
to  be  in  a  straight  line,  as  viewed  from  E.  Thus  in 
the  complete  circuit,  the  average  apparent  speeds  of 
the  sun  and  c,  as  seen  from  the  earth,  are  the  same; 
but  in  the  upper  half  of  the  path,  as  shown  in  the 
diagram,  the  sun  will  appear  to  be  moving  faster  than 
c,  in  the  lower  half  it  will  appear  to  be  moving  slower. 
The  apparent  variation  in  the  speed  of  the  sun  as  it 
revolves  about  the  earth  can,  therefore,  be  explained 
as  the  resultant  of  two  component  motions,  each  uni- 
form and  circular.  The  larger  circle,  in  the  terms  of 
the  ancient  mathematicians,  is  the  deferent,  the  smaller 
circle,  the  epicycle. 

By  varying  the  relative  sizes  of  the  two  circles  and 
the  respective  speeds  of  revolution  in  each,  very  dif- 
ferent types  of  resultant  motion  for  the  sun  may  be 
obtained.  In  fact,  by  a  combination  of  a  number  of 
epicycles,  by  piling 

"Cycle  upon  cycle,  orb  on  orb," 

practically  any  irregular  motion  can  be  analyzed,  or 
resolved,  into  a  corresponding  number  of  uniform  cir- 
cular motions. 

The  epicycle  is  not  only  a  mathematical,  it  is  also 
a  mechanical,  device,  being  used  in  many  types  of 
machines.  As  used  by  Hipparchus  and  his  modern 
successors  to  describe  the  motions  of  the  planets,  it 
is  a  mathematical  device,  purely  and  simply,  a  com- 


The  Law  of  Gravitation          73 

puter's  fiction  for  keeping  track  of  irregular  motions, 
and  did  not  and  does  not  involve  any  idea  as  to  the 
actual  construction  of  the  planetary  system. 

From  and  after  the  time  of  Hipparchus  the  varied 
motions  of  the  planets  were  carefully  studied,  the 
crude  tables  of  their  motions  gradually  perfected  and 
simplified.  The  development  was  extremely  slow,  each 
step  in  the  process  requiring  centuries  of  effort.  A 
few  great  names  stand  out — Ptolemy,  Copernicus, 
Tyco  and  Kepler.  Ptolemy  was  the  direct  successor 
of  Hipparchus;  he  elaborated,  extended,  and  perfected 
the  methods  of  that  more  illustrious  astronomer.  He 
welded  together  all  the  observations  and  theories  of 
his  time  into  a  compact  and  consistent  theory  of  the 
universe;  he  formed  and  gave  to  the  world  in  the 
year  150  A.D.  a  complete  digest  of  astronomy,  a  digest 
so  complete,  so  consistent,  that  it  remained  for  over 
thirteen  centuries  the  standard  of  scientific  knowledge. 
He  knew  that  the  earth,  taken  as  a  whole,  is  a  sphere, 
and  he  knew  its  approximate  size ;  his  tables  represented 
the  motions  of  the  sun,  moon,  and  planets  with  con- 
siderable accuracy,  an  accuracy  truly  remarkable  for 
an  age  without  means  of  accurate  observations. 
Ptolemy,  unfortunately,  fell  into  the  not  unnatural 
error  of  thinking  the  earth  to  be  at  the  centre  of  the 
universe,  and  the  sun,  the  planets,  and  the  stars  as 
minor  bodies  revolving  about  it.  This,  of  course, 
introduced  complications  into  his  mathematical  tables, 


74     Gravitation  versus  Relativity 

but  to  him  such  complications  seemed  of  less  moment 
than  the  physical  difficulties  of  a  rotating  earth.  His 
cardinal  error  was  in  failing  to  recognize  the  atmos- 
phere as  a  part  of  the  earth  and  as  partaking  in  any 
motions  of  the  earth :  to  him  a  rotation  of  the  earth 
from  west  to  east  would  give  rise  to  terrific  winds, 
blowing  at  the  rate  of  nearly  a  thousand  miles  per 
hour,  hurricanes  which  would  sweep  the  earth  bare  of 
trees  and  houses.  It  was  for  this  physical  reason  that 
Ptolemy,  disregarding  the  hints  and  suggestions  of 
former  writers,  adopted  his  system  of  an  immovable, 
non-rotating  earth,  with  its  unnecessary  mathematical 
complications. 

Not  until  the  sixteenth  century  was  there  any  radical 
change  in  the  methods  and  theories  of  Ptolemy. 
Numerical  corrections  had  been  made,  and  his  tables 
enlarged  and  improved;  but  fundamentally  no  changes 
had  been  made  in  the  Ptolemaic  system.  In  the  popu- 
lar mind  the  idea  of  the  epicycle  became  fixed  and 
the  universe  was  thought  of  as  consisting  of  a  com- 
plicated system  of  revolving  circles  and  spheres.  The 
beauty  of  the  purely  mathematical  computing  device 
was  lost  in  the  attempt  to  realize  the  mechanism  of 
the  universe.  The  first  great  advance  was  made  by 
Copernicus,  whose  book  was  published  in  1543.  He 
was  struck  with  the  unnecessary  intricacies  of  the 
Ptolemaic  system,  with  the  utter  impossibility  of  the 
innumerable  stars  revolving  about  the  earth  every 


The  Law  of  Gravitation  75 

twenty-four  hours.  Through  travel  and  voyages  to 
distant  lands  a  truer  knowledge  of  the  earth  and  the 
atmosphere  had  gradually  developed,  and  Copernicus 
realized  that  the  physical  objections  of  Ptolemy  to  a 
rotating  earth  were  not  valid.  He  revived  the  older 
conception  of  Nicetas  of  a  rotating  earth,  and  of 
Pythagoras  of  a  central  sun,  and  combined  them  into 
a  simple  and  beautiful  system,  which  fully  explained 
the  larger  motions  of  the  sun  and  planets.  In  this 
system  the  sun  is  placed  at  the  centre  of  the  universe, 
and  around  it  revolve  the  planets  in  circular  paths, 
Mercury,  Venus,  the  earth,  Mars,  etc. :  the  moon  being 
recognized  as  a  secondary  body  revolving  about  the 
earth.  This  approximates  very  closely  to  the  true 
system  of  the  planets.  Copernicus,  himself,  realized 
that  there  are  irregularities  in  the  motions  of  the  sun 
and  planets,  which  his  system,  by  itself,  cannot  explain, 
and  he  was  forced  to  introduce  a  certain  number  of 
epicycles.  By  his  rearrangement  he  struck  out  the 
larger  epicycles  of  Ptolemy  and  reduced  the  number, 
but  he  still  retained  their  machinery  to  explain,  or 
account  for,  the  smaller  motions  of  the  planets. 

Before  further  advances  could  be  made  in  the 
theoretical  explanation  of  the  planetary  motions,  these 
motions  themselves  needed  a  thorough  investigation 
and  a  careful  tabulation.  For  more  than  fourteen  cen- 
turies there  had  been  no  improvement  in  the  art  of 
observing;  the  instruments  of  Hipparchus  and  Ptolemy 


76     Gravitation  versus  Relativity 

were  still  in  use.  The  time  was  ripe  for  a  new 
astronomer,  one  who  would  devote  his  entire  lifetime 
to  a  diligent  study  of  the  heavens,  one  who  would  take 
no  fact,  no  motion,  on  the  authority  of  the  ancients, 
but  who  would  determine  everything  for  himself. 
New  instruments  were  to  be  invented,  new  methods 
of  observing  to  be  discovered.  An  observer,  not  a 
mathematician,  was  needed.  This  was  the  place  filled 
by  Tycho  Brahe ;  he  supplied  the  materials  from  which 
his  successors  built  the  structure  of  modern  astronomy. 

Tycho  died  in  1601  and  left  his  uncompleted  manu- 
scripts and  his  observations  to  his  student  and  fol- 
lower— Kepler.  Eight  years  later  this  illustrious 
astronomer  announced  his  first  two  laws  of  planetary 
motion,  and  completed  his  discoveries  nine  years  there- 
after with  the  famous  third  law.  In  all  he  had  devoted 
twenty-two  years  to  his  search,  but  he  had  solved  the 
problem  of  planetary  motions.  By  his  discovery  of 
these  laws  he  swept  aside  all  the  old  theories  and 
machinery  of  the  heavens,  and  laid  the  foundation 
upon  which  Newton  built  the  wonderful  edifice  of 
universal  gravitation. 

These  laws  of  Kepler  summarized  the  observations 
of  centuries,  from  the  crude  measures  of  the  Chaldeans 
to  the  refined  and  then  unsurpassed  observations  of 
Tycho.  They  are  so  fundamental  to  a  clear  under- 
standing of  the  real  motions  of  the  planets  and  of 
the  Newtonian  law,  that  a  full  explanation  is  necessary 


The  Law  of  Gravitation          77 

as  to  exactly  what  these  laws  are  and  as  to  what  they 
actually  mean.  The  first  two  laws  describe  the  motions 
of  a  planet  about  the  sun;  the  third  law  establishes 
a  relationship  between  the  orbits  of  the  different  planets. 
The  three  are : 

1.  The  path  of  each  planet  is  an  ellipse,  the  sun 
being  at  one  focus. 

2.  The  line  joining  the  sun  and  the  planet  sweeps 
over  equal  areas  of  space  in  equal  intervals  of 
time. 

3.  The   squares   of  the  periodic  times  of  two 
planets  (lengths  of  their  respective  years)  are 
proportional  to  the  cubes  of  their  mean  dis- 
tances from  the  sun.    (One-half  of  the  sum  of 
the  greatest  and  least  distances.) 

The  exact  meaning  of  these  three  laws  is  shown  in 
the  annexed  figure: 

A  planet,  Mercury  for  instance,  travels  about  the 
sun,  S,  in  the  mathematical  curve  commonly  called  an 
oval,  but  technically  known  as  an  ellipse,  and  shown 
in  the  diagram  as  ACBD.  This  curve  is  not  placed 
symmetrically  with  respect  to  the  sun;  it  is  off  centre, 
so  to  speak,  and  the  sun  lies  at  a  point  called  the  focus. 
In  this  curve  the  planet  travels  at  varying  speeds; 
when  near  the  sun  it  travels  considerably  faster  than 
when  at  a  more  distant  part  of  its  path,  and  this 
variation  in  speed  is  the  subject  of  Kepler's  second 
law.  In  a  given  interval  of  time,  one  week  for  example, 
the  planet  will  travel  from  A  to  C,  and  the  line  join- 


78     Gravitation  versus  Relativity 

ing  it  to  the  sun  will,  during  this  interval,  sweep  over 
the  area  ASC.  Some  time  later,  in  one  week  the 
planet  will  travel  from  B  to  D  and  Kepler's  law  states 


Fig.  ii.    Kepler's  Laws  of  Planetary  Motion. 

that  the  distance  thus  travelled  is  so  proportioned  to 
the  former  distance  AC,  that  the  area  SBD  is  equal 
to  the  area  ASC. 

These  two  laws  fully  describe  the  motion  of  the 
planet,  and  enable  one  to  keep  track  of  its  movements 
in  the  heavens  and  to  predict  where  it  may  be  seen 
at  a  given  time.  They  did  away  with  the  need  of 
epicycles;  the  irregular  motion  of  the  planet,  its  varia- 
tion in  speed  and  direction,  were  fully  accounted  for 
by  the  shape  of  the  path  and  the  varying  speed  of  the 
planet  in  that  path. 

The  third  law  expresses  a  relation  between  the  orbits 


The  Law  of  Gravitation  79 

of  two  planets;  between  the  size  of  the  orbits  and  the 
length  of  time  required  for  each  planet  to  complete 
one  trip  around  its  oval  path. 

Now  these  laws  of  Kepler  sufficed  to  trace  fully  the 
paths  of  the  planets,  as  shown  by  the  observations  of 
that  date.  The  telescope  was  not  then  invented,  and 
the  observations  were  still  crude  and  approximate  in 
comparison  with  modern  accuracy.  Irregularities  of 
motion,  too  small  to  be  detected  in  the  observations 
of  Tyco,  are  now  known  to  exist,  and  the  laws  of 
Kepler  are  now  known  to  be  but  approximations,  but 
remarkably  close  and  accurate  approximations. 

While  this  line  of  eminent  astronomers  were  study- 
ing the  heavens  and  analyzing  the  varied  motions  of 
the  planets,  other  scientists  were  studying  the  laws  of 
terrestrial  mechanics  and  inventing  methods  of  research 
and  calculation.  The  device  of  logarithmic  computa- 
tion doubled  or  quadrupled  the  effective  working  life 
of  an  astronomer,  and  allowed  computations  to  be 
made,  which  otherwise  would  have  been  hopeless  from 
their  very  length  and  complexity.  In  the  realm  of  the 
physical  sciences  the  name  of  Galileo  stands  pre- 
eminent. He  laid  the  foundation  of  the  modern  science 
of  mechanics,  he  formulated  the  fundamental  laws  of 
motion,  discovered  the  law  of  acceleration  of  falling 
bodies,  and  determined  the  path  of  a  projectile.  He 
invented  the  telescope  as  an  instrument  of  astronomical 
research,  and,  by  its  use,  discovered  many  facts  that 


8o     Gravitation  versus  Relativity 

reinforced  and  proved  the  general  correctness  of  the 
Copernician  system — the  rotation  of  the  sun  on  its 
axis,  the  phases  of  the  planets,  and  the  satellites  of 
Jupiter. 

Newton,  the  greatest,  the  most  eminent  scientist  of 
all  ages,  brought  together  the  physical  discoveries  of 
Galileo  and  the  astronomical  laws  of  Kepler,  and  united 
the  whole  mass  of  unrelated  laws  and  phenomena  into 
a  single,  all  comprehensive,  fundamental  law — the  law 
of  universal  gravitation.  The  laws  of  Kepler  were 
shown  by  Newton  to  be  the  inevitable  result  of  a 
force  of  attraction,  directed  towards  the  centre  of  the 
sun  and  diminishing  in  intensity  proportionally  as 
the  square  of  the  distance  therefrom  increased.  This 
force  keeps  the  planets  in  their  orbits  and  directs  their 
movements.  Reasoning  by  analogy,  it  was  easy  for 
Newton  to  infer  that  the  moon  was  kept  in  her  orbit 
by  a  similar  force  directed  towards  the  centre  of  the 
earth.  Bodies  on  the  surface  of  the  earth  fall  freely, 
acted  upon  by  a  similar  force.  But  are  the  two  forces, 
the  force  that  keeps  the  moon  in  her  distant  path  and 
the  force  that  causes  a  stone  to  fall,  are  these  two 
forces  one  and  the  same?  Newton  thought  so,  and 
in  1665  tried  to  prove  his  hypothesis.  The  stone  falls 
towards  the  earth  16.095  ^ee^  m  a  single  second  of 
time.  And  if  the  force,  which  causes  it  to  fall,  de- 
creases as  the  square  of  the  distance  from  the  earth's 
centre,  then  at  a  distance  of  60.3  times  the  radius  of 


The  Law  of  Gravitation          81 

the  earth  the  stone  would  fall  only  1/3636^1  as  far. 
At  this  distance,  which  is  the  average  distance  of  the 
moon  from  the  earth's  centre,  the  stone  would,  there- 
fore, fall  only  about  one-twentieth  of  an  inch,  actually 
0.0532  inches. 

Now,  if  the  force  that  keeps  the  moon  in  her  orbit 
is  the  same  as  that  acting  on  the  stone,  then  the  moon 
should  be  falling  towards  the  earth  by  exactly  this 
same  amount,  namely  0.0532  inches,  in  one  second. 
The  amount  that  the  moon  actually  does  fall  is  the 
amount  by  which  it  is  deflected  from  a  straight  line, 
as  it  travels  in  its  orbit  and  as  shown  in  Figure  12. 

If  it  were  not  for  the  attraction  of  the  earth,  the 
moon  would  move  in  the  straight  line  indicated  by  the 
arrow  AD.  The  pull  of  the  earth,  however,  causes 
the  moon  to  travel  in  the  curved  orbit  AB.  In  one 
second  of  time  the  moon  moves  over  the  arc  AB,  and 
consequently  falls  towards  the  centre  of  the  earth  by 
a  distance  represented  by  the  line  AC.  This  distance 
can  be  calculated  as  soon  as  the  length  of  the  arc  AB 
is  known  together  with  the  actual  size  of  the  moon's 
orbit.*  The  angular  value  of  this  arc  can  readily  be 
computed,  for  it  takes  the  moon  some  27  days  7  hours 
and  43.2  minutes  to  make  one  complete  revolution 

*  Considering  the  arc  AB  as  a  straight  line  and  the  triangle 
ABF  as  right  angled,  we  have 

AC   :  AB  =  AB   :  AF  (or  2r) 
Ar        (AB)a 


A 

whence:  A 


82      Gravitation  versus  Relativity 


about  the  earth.  Reducing  this  to  seconds,  the  sidereal 
period  of  the  moon  is  2,360,591  seconds  and,  there- 
fore, the  arc  AB  is  1/2,360,59 ith  part  of  a  circum- 


Fig.  12.    The  Fall  of  the  Moon  towards 
the  Earth. 

ference.  On  the  other  hand,  to  find  the  size  of  the 
moon's  orbit  the  actual  dimensions  of  the  earth  in 
feet  must  be  known,  for  the  radius  of  the  orbit  is 
60.3  times  the  radius  of  the  earth.  In  1665  the  exact 
size  of  the  earth  was  not  known,  the  length  of  a  degree 
on  the  surface  was  supposed  to  be  sixty  (60)  miles, 


The  Law  of  Gravitation  83 

which  made  the  radius  of  the  earth  18,150,000  feet. 
With  these  figures  the  distance  AC  through  which 
the  moon  falls  in  a  single  second  comes  out  as  0.0457 
inches,  which  differs  by  about  15%  from  the  0.0532 
inches  found  for  a  stone  at  the  moon's  distance  from 
the  earth. 

This  difference  of  15%,  or  a  trifle  less  than  one  one- 
hundred  and  thirtieth  (i/i3Oth)  of  an  inch  in  the 
motion  of  the  moon,  caused  Newton  to  consider  his 
theory  as  not  proved  and  he  laid  aside  his  work.  Nearly 
twenty  years  later  the  measurements  of  Picard,  on 
the  length  of  a  meridian  arc,  were  brought  to  New- 
ton's attention  by  Halley,  who  urged  him  to  take  up 
again  his  discarded  work  and  to  revise  his  calculations. 
These  measurements  showed  one  degree  to  be  some- 
what over  69  miles  in  length,  instead  of  the  even  60 
used  by  Newton  in  his  former  calculations.  The  true 
length  of  the  radius  of  the  earth  was  thus  shown  to 
be  nearly  twenty-one  million  feet  (20,926,000)  instead 
of  the  eighteen  million  used  by  Newton  in  his  first 
calculations.  With  these  new  figures,  the  distance 
that  the  moon  falls  towards  the  earth  comes  out  as 
0.0535  inches,  which  agrees  with  the  figure  derived 
from  the  falling  stone  to  within  less  than  one  one- 
thousandth  of  an  inch  (3/10,000),  or  to  within  three- 
quarters  of  one  per  cent. 

Thus  Newton  in  1685  proved  that  the  force,  which 
retains  the  moon  in  her  orbit,  is  the  same  force,  which 


84     Gravitation  versus  Relativity 

causes  bodies  to  fall  to  the  earth,  and  that  this  force, 
gravitation,  varies  inversely  as  the  square  of  the  dis- 
tance from  the  earth's  centre.  The  sun  attracts  the 
earth  and  its  satellite,  the  earth  attracts  the  moon,  and, 
unless  the  attraction  of  the  earth  ceases  at  some  point, 
the  earth  must  also  attract  the  sun  and  all  the  planets 
as  well.  Thus  the  earth  attracts  each  and  every  particle 
of  matter,  the  sun  attracts  each  and  every  planet  and 
satellite,  and  in  turn  the  planets  and  satellites  attract 
the  sun.  Thus  did  Newton  infer  the  law  of  universal 
gravitation. 

The  law  of  gravitation  is  thus  an  empirical  law; 
it  is  deduced  from  experience.  So  far  as  is  known 
to-day,  gravitation  is  a  fundamental  property  of  mat- 
ter. The  action  of  the  force  is  instantaneous,  it  is  not 
modified,  in  any  degree,  by  the  interposition  of  other 
particles  of  matter,  and  it  is  independent  of  the  struc- 
ture and  condition  of  a  body :  it  is  the  same  for  ice, 
water,  and  steam. 

As  enunciated  by  Newton,  this  law  deals  with  indi- 
vidual particles  of  matter,  separated  by  finite  dis- 
tances. It  does  not  deal  with  forces  of  cohesion,  or 
with  the  molecular  forces  which  may  determine  the 
physical  condition  of  a  body.  For  separate  particles 
of  matter  the  mathematical  statement  of  the  law  is 
very  simple. 

Particle  A,  for  example,  attracts  particles  B,  C, 
and  D,  and  is  in  turn  attracted  by  each  and  every  one 


The  Law  of  Gravitation          85 

of  those  particles.  The  amount  of  this  attraction 
depends  in  each  case  upon  two  factors ;  the  product  of 
the  masses  of  the  two  particles  under  consideration, 


Fig.  13.    Mutual  Attractions  of  Particles  of 
Matter. 

and  their  distance  apart.  Thus  the  force  between  A 
and  B  is  measured  by  the  mass  of  A  multiplied  by 
that  of  B,  and  the  product  divided  by  the  square  of 
the  distance  between  A  and  B.  At  double  the  distance 
apart  the  force  is  one-quarter,  at  ten  times  the  distance, 
one  one-hundredth.  And  this  law  holds  for  individual 
particles  no  matter  what  the  distance  between  A  and 

B,  whether  measured  by  the  thousandth  of  an  inch, 
or  by  millions  of  miles. 

But  a  body  of  material  size  is  made  up  of  a  collection 
of  particles,  more  or  less  firmly  bound  together.  That 
is,  if  the  particles  B,  C,  and  D  are  brought  together  at 

C,  they  might  be  considered  as  forming  a  single  body, 
but  as  these  three  particles  cannot  occupy  the  same  space 
at  the  same  time,  the  actual  distances  of  the  three  from 
A  will  still  differ  by  minute  quantities,  and  the  total 


86      Gravitation  versus  Relativity 

force  acting  upon  A  must  still  be  determined  by  taking 
into  account  these  differing  distances.  Thus,  while  the 
mathematical  expression  of  the  law  is  very  simple  when 
stated  in  terms  of  particles,  it  becomes  complicated 
when  applied  to  actual  material  bodies.  In  fact,  it 
becomes  impossible,  except  in  one  or  two  simple  cases, 
to  find  the  exact  mathematical  expression  for  the  attrac- 
tion between  two  bodies. 

The  great  exception  is  for  bodies  of  a  spherical  form. 
The  force  of  attraction  between  two  homogeneous 
spheres  of  matter,  two  balls  of  iron  or  of  copper,  can  be 
expressed  with  the  simplicity  of  that  for  two  particles. 
Such  spherical  balls  of  matter  act  as  though  their  entire 
masses  were  concentrated  at  their  respective  centres : — 
the  only  distance  that  enters  the  formula  is  the  distance 
between  the  two  centres.  Similarly  for  a  shell  of 
spherical  form;  it  acts  as  if  all  its  constituent  particles 
were  united  into  a  single  particle  at  the  centre.  Thus  a 
body  made  up  of  a  series  of  spherical  shells  of  different 
densities,  or  even  of  different  materials,  acts  as  though 
the  entire  mass  were  concentrated  at  the  centre,  the 
only  condition  being  that  each  shell  be  uniform  in  itself. 
That  is,  the  law  of  attraction  can  be  very  simply,  but 
rigorously,  expressed  for  solid  spheres  of  copper,  silver, 
or  lead;  for  thin  spherical  copper  vessels  filled  with 
water,  or  with  melted  lead;  for  a  sphere  made  up  of 
concentric  layers  of  copper,  iron,  gold,  and  lead.  It 
makes  no  difference  whether  the  denser  material  be 


The  Law  of  Gravitation  87 

inside  or  outside,   provided  only  that  each   spherical 
layer  be  uniform. 

If,  however,  the  shape  of  the  ball  be  changed  to  an 
ellipsoid,  or  egg-shaped  body,  then  at  once  the  simple 
expression  for  the  attraction  vanishes,  and  is  replaced 
by  a  most  complicated  formula.  If  the  body  be  non- 
symmetrical,  or  irregularly  shaped,  then  it  becomes  im- 
possible to  find  a  complete  mathematical  expression  for 
the  force  of  attraction  it  exerts  upon  another  body. 
This  force,  of  course,  is  a  certain  definite  quantity, 
but  we  cannot  express  that  quantity  in  definite  mathe- 
matical language.  In  other  words,  if  we  have  two 
bodies,  both  made  of  copper  of  uniform  density,  one 
a  perfect  sphere  two  feet  in  diameter  and  the  other  an 
irregular  lump,  we  can  compute  exactly  the  attractive 


Fig.   14.     Attractions  of  a  Sphere  and  of  an 
Irregular  Body 

force  of  the  sphere,  A,  but  we  cannot,  except  very 

roughly,  determine  mathematically  that  of  the  lump,  B. 

An  analogous,  but  not  exactly  similar,  case  might  be 


88     Gravitation  versus  Relativity 

used  as  an  illustration.  If  we  have  a  sphere,  or  a  cube, 
of  copper  of  known  density,  we  can  readily  measure 
with  calipers  the  diameter  of  the  sphere  or  the  side  of 
the  cube,  and  then  by  perfectly  simple  mathematical 
formulas  we  can  compute  the  volume  of  the  sphere  or 
cube,  and  then  from  the  known  weight  of  a  cubic  inch 
of  copper  we  can  calculate  the  exact  weight  of  the  body. 
But  if  we  have  a  lump  of  copper,  irregular  in  shape, 
with  excresences  sticking  out  in  all  directions,  it  would 
be  well-nigh  impossible,  by  any  system  of  measure- 
ments, to  determine  its  exact  volume  and  weight.  Of 
course,  in  this  case  the  weight  of  the  lump  could  be  at 
once  determined  by  a  pair  of  scales,  but  not  by  mathe- 
matical calculation. 

The  sphere  is  the  only  type  of  celestial  body  of  which 
the  force  of  attraction  can  be  rigorously  calculated. 
For  bodies  of  any  other  shape,  approximations  must  be 
used. 

There  is  a  large  class  of  bodies,  variously  known  as 
ellipsoids,  or  spheroids,  some  of  which  differ  very  little 
from  spheres,  and  this  class  is  of  importance  in  the 
discussion  of  the  planetary  motions.  The  important 
type  of  bodies  are  those  formed  by  the  revolution  of 
an  ellipse  about  one  of  its  axes.  If  in  the  figure  the 
ellipse  ANBS  be  revolved  about  the  axis  NS,  it  will 
develop  a  solid,  such  that  every  section  perpendicular 
to  this  axis  will  be  a  circle,  every  section  through  the 
axis  will  be  an  ellipse  exactly  similar  to  the  one  shown, 


The  Law  of  Gravitation 


89 


and  every  other  section,  an  ellipse  intermediate  between 
the  circle  and  ANBS.  The  small  circle,  inscribed  in  the 
ellipse,  will  develop  a  sphere.  Now,  it  is  a  very  simple 


Fig.  15.    Attractions  of  Spheres  and  Spheroids. 

problem  in  geometry  to  find  the  volumes  of  the  sphere 
and  of  the  ellipsoid.  If  we  know  the  density  of  the 
material,  we  can  find  the  total  amount  of  matter  in  the 
spheroid  just  as  easily  as  we  can  that  in  the  sphere, 
and  we  can  then  find  the  respective  weights  of  these 
bodies,  with  an  equal  degree  of  accuracy  in  each  case. 
A  sphere,  shown  by  the  dotted  line,  can  also  be  found, 
such  that  its  mass,  or  weight,  is  exactly  equal  to  that 
of  the  spheroid,  M. 

But  the  case  is  different,  when  the  attractions  of  these 
bodies  upon  a  particle  of  matter  at  P  is  required.  Each 
of  the  spheres  attract  P  as  though  their  entire  masses 


90      Gravitation  versus  Relativity 

were  concentrated  at  the  centre,  c.  If  the  mass  of  the 
particle,  P,  be  taken  as  unity,  the  attraction  of  the 
equivalent  sphere  is 

M 


But  it  is  clear  from  the  figure  that  this  is  not  so  for  the 
spheroid,  for  quite  a  bulge  of  matter  at  B  is  nearer  to 
P  than  any  of  the  matter  in  the  sphere,  and  a  similar 
bulge  at  A  is  farther  away.  While  these  bulges  are 
symmetrically  placed  as  regards  the  sphere,  yet  that  at 
B  is  much  nearer  to  P  than  that  at  A,  and  will,  there- 
fore attract  P  more  strongly,  and  so  it  is  easy  to  see 
that,  as  a  whole,  the  spheroid  will  attract  P  more 
strongly  than  will  the  equivalent  sphere.  The  amount 
of  this  excess  attraction  over  that  of  the  equivalent 
sphere  depends  upon  two  factors ;  first  the  shape  of  the 
spheroid,  upon  its  eccentricity  in  other  words,  and  sec- 
ond, the  relative  distance  of  P  from  the  centre  as  com- 
pared to  the  size  of  the  spheroid,  or  upon  the  ratio  of 
CB  to  CP. 

If  the  point  P,  instead  of  being  in  what  might  be 
called  the  equator  of  the  spheroid,  were  in  the  prolonga- 
tion of  the  axis,  or  directly  over  what  might  be  called 
the  north  pole,  then  it  is  easy  to  see  from  the  figure  that 
the  equivalent  sphere  would  attract  P  more  strongly 
than  the  ellipsoid.  The  attraction  is  the  least  when  P 
is  in  the  prolongation  of  the  axis,  greatest  when  in  the 


The  Law  of  Gravitation          91 

so-called  equator.  At  some  point  midway  between 
these  two,  the  attraction  of  the  spheroid  will  exactly 
equal  that  of  the  equivalent  sphere.  This  point  is  54° 
44'  from  the  axis.  The  attraction  of  the  spheroid  upon 
P  varies,  therefore,  not  only  with  the  distance  of  P 
from  the  centre,  but  also  with  the  direction  of  P  with 
respect  to  the  axis  of  rotation.  It  is  thus  impossible  to 
find  a  simple  and  accurate  mathematical  expression  for 
the  attraction  of  the  spheroid  upon  a  particle  in  space. 
The  very  best  that  can  be  done  is  to  express  this  force 
in  a  series  of  terms,  the  first  of  which  is  the  attraction 
of  the  equivalent  sphere,  the  following  terms  growing 
smaller  and  smaller.  This  expression  is 


in  which  e  is  the  ellipticity  of  the  spheroid  and  Y> 
the  angular  distance  of  the  particle  P  from  the  axis 
NS.  To  fully  appreciate  the  meaning  of  this  formula, 
consider  the  specific  case  in  which  the  shorter  axis  of 
the  figure  is  just  one-half  the  longer,  and  find  the  attrac- 
tion of  the  spheroid  upon  particles,  distant  from  the 
centre  just  twice  the  longer  axis  of  the  body.  In  this 
case  s  is  equal  to  l/2t  as  is  also  the  ratio  a/r.  For  a 
particle,  P',  in  the  equator  of  the  body  y  is  90°,  and 
cos  Y  is  zero,  and  the  expression  reduces  to: 

M 


MT  .  Axi 

r*  I1   h  10  X  4 


92     Gravitation  versus  Relativity 

and  the  whole   attraction   of  the   spheroid  upon   P' 
becomes, 


that  of  the  equivalent  sphere. 

Similarly  for  a  particle,  P"'  above  the  north  pole, 
the  expression  becomes: 

M 


Mr       6      ii 

?  I1 ""  m  x  4] 


or 


40 

that  of  the  equivalent  sphere. 

The  particle  at  P"  will  be  attracted  with  a  force 
exactly  equal  to  that  of  the  sphere.  Thus  it  is  seen 
that  for  particles  situated  at  the  same  distance  from 
the  centre  of  the  ellipsoidal  body,  the  force  of  attraction 
varies  by  9/4Oths  of  its  total  amount,  by  22  l/2%, 
varying  thus  with  mere  changes  in  direction.  These 
results,  however,  are  not  fully  accurate,  for  only  one 
term  of  the  expression  for  the  force  has  been  taken  ac- 
count of:  there  are  in  reality  many  other  terms,  but 
these  are  all  small  in  comparison  with  the  one  used. 

But  as  the  distance  of  the  particle  from  the  body 
increases,  the  total  attraction  approaches  more  and 
more  closely  to  that  of  the  equivalent  sphere.  This 


The  Law  of  Gravitation          93 

approach  is  very  rapid,  for  the  expression  for  the 
force  depends  upon  the  square  of  the  ratio  a/r.  In  the 
above  special  example,  if  the  particle  were  removed  to 
a  distance  equal  to  ten  times  the  longer  axis  of  the 
spheroid,  then  the  force  of  attraction  of  the  spheroid 
would  differ  by  less  than  ^2  of  one  per  cent  from  that 
of  the  sphere  of  equal  mass :  at  a  distance  of  one  hun- 
dred (100)  radii,  this  difference  of  attractions  for  the 
sphere  and  the  spheroid  would  be  considerably  less  than 
one  one-hundredth  of  one  percent. 

If  it  is  thus  impossible  to  find  a  rigorous  mathe- 
matical expression  for  the  attraction  of  such  a  simple 
body  as  an  ellipsoid,  how  utterly  impossible  must  it 
be  to  find  any  expression  for  the  attraction  of  an 
irregularly  shaped  body!  Approximations  must  be 
used,  and  the  first,  the  FUNDAMENTAL  APPROXIMATION 
used  in  all  astronomical  researches  since  Newton  an- 
nounced the  law  of  gravitation,  is  the  assumption  that 
all  the  bodies  of  the  solar  system  are  homogeneous 
spheres. 

This  assumption  is  necessary,  it  cannot  be  avoided. 
It  may  approximate  very  closely  to  the  truth,  but  it  is 
an  approximation,  and  any  motions,  or  laws  of  motion, 
derived  by  the  use  of  this  assumption  are  necessarily 
approximations. 

Fortunately  there  are  two  circumstances,  which  make 
this  approximation  allowable  in  treating  of  the  motions 
of  the  planets : — the  bodies  of  the  solar  system  differ 


94     Gravitation  versus  Relativity 

but  little  from  spheres,  and  their  mutual  distances  apart 
are  very  great  as  compared  to  their  individual  di- 
mensions. Saturn  differs  more  greatly  from  a  sphere 
than  any  other  known  body  of  the  system.  Viewed  in 
a  telescope,  its  disc  is  noticeably  elliptical,  and,  in  fact, 
its  ellipticity  is  found  to  be  almost  exactly  i/9th. 
Jupiter  also  shows  a  disc  distinctly  not  circular,  and 
measurements  indicate  its  ellipticity  to  be  i/i7th.  All 
the  other  bodies  approach  much  more  closely  to  true 
spherical  forms;  the  ellipticity  of  the  earth  is  but 
i/293th  while  that  of  the  sun  is  extremely  minute. 
But,  on  the  other  hand,  the  distances  between  these 
bodies  are  relatively  very  great.  At  her  nearest 
approach  to  the  earth,  Venus  is  distant  over  six  thou- 
sand times  the  radius  of  the  earth,  and  the  excess 
attraction  due  to  the  ellipticity  of  the  earth  amounts  to 
only  about  one  twelve-billionth  that  of  the  whole  force 
beween  the  two  bodies. 

It  is  different,  however,  when  the  motions  of  the 
satellites  are  considered.  As  early  as  1748  Euler 
showed  that  the  spheroidal  figure  of  Jupiter  would 
cause  irregularities  in  the  motions  of  his  satellites,  and 
ten  years  later  Walmsley  showed  that  the  elliptical 
shape  of  Jupiter  would  cause  a  rotation  of  the  orbit  of 
each  satellite,  a  rotation  exactly  similar  to  the  now  much 
discussed  motion  of  the  perihelion  of  Mercury.  The 
ellipticity  of  the  earth  affects  the  motion  of  the  moon 
to  a  very  noticeable  amount,  and  such  ellipticity  is  taken 


The  Law  of  Gravitation          95 

account  of  in  all  theories  of  the  moon's  motion  and  in 
all  tables  from  which  her  place  in  the  heavens  is 
predicted. 

The  planets,  as  has  been  seen,  are  so  small  relative 
to  their  mutual  distances  apart,  that  their  individual 
shapes  can  have  no  appreciable  effect  upon  their  motions. 
Not  so,  however,  with  the  sun ;  its  dimensions  are  ap- 
preciable fractions  of  the  planetary  distances.  Mercury 
is  distant  only  eighty  (80)  times  the  solar  radius,  the 
earth  two  hundred  and  fifteen  (215).  The  distance 
factor  in  the  expression  for  the  excess  attraction  due  to 
the  shape  of  the  central  body  is  thus  for  the  case  of 
Mercury  1/6400;  for  the  earth  1/46,000.  If,  therefore, 
the  sun  be  not  strictly  spherical,  the  variation  in  the 
force  of  gravitation  due  to  its  shape  may  have  a 
measurable  effect  upon  the  motions  of  the  nearer 
planets. 

In  all  the  planetary  theories  and  computations  the  sun 
has  been  regarded  as  a  sphere,  and  all  influences  due 
to  a  possible  departure  from  such  shape  have  been 
omitted.  From  the  time  Galileo  first  turned  his  puny 
telescope  on  the  sun  and  discovered  its  surface  covered 
with  dark  ' 'spots, "  it  has  been  known  that  the  sun's 
density  is  not  uniform;  from  the  time  Newton  evolved 
his  law  in  1686,  it  has  been  recognized  that  the  sun 
is  not  a  sphere  of  uniform  density,  and  that,  in  neglect- 
ing all  questions  as  to  its  true  shape  and  condition,  a 
fundamental  and  far-reaching  approximation  was  used. 


96     Gravitation  versus  Relativity 

In  the  earlier  days  of  celestial  mechanics,  when  observa- 
tions were  still  crude,  the  use  of  this  approximation  was 
fully  justified,  whether  it  be  so  today  is  a  moot  question. 
During  the  last  century  numerous  series  of  measures 
have  been  made  to  determine,  if  possible,  the  exact 
shape  of  the  sun. 

Every  series  of  measures,  heretofore  made,  shows 
a  distinct,  measurable  departure  from  sphericity,  but 
these  departures  are  extremely  minute.  Such  measures 
are  extremely  difficult  to  make,  for  the  visible  edge  of 
the  sun  is  not  a  distinct,  clean-cut  line;  it  is  hazy,  in- 
definite, and  fades  out  gradually.  Further,  the  atmos- 
phere of  the  earth,  through  which  the  light  comes  from 
the  sun,  is  seldom  in  a  state  of  absolute  rest,  and  the 
movements  of  the  air  currents  cause  apparent  trem- 
blings or  "boilings"  of  the  telescopic  images  of  sun  and 
stars.  This  unsteadiness  of  the  atmosphere  is  more 
marked  by  day  than  at  night;  this  being  due  probably 
to  the  heating  effects  of  the  sun's  rays.  Photographs 
of  the  sun  are  made  with  exposures  of  about  i/ioooth 
of  a  second,  yet  even  on  such  photographs  the  ' 'boiling" 
is  noticeable,  and  only  at  rare  intervals  will  a  perfect 
plate  be  obtained. 

During  the  first  half  of  the  nineteenth  century  there 

•were  many  visual  observations,  or  measurements  of  the 

size  and  shape  of  the  sun.    The  results  were  not  very 

accordant,  differences  in  the  lengths  of  the  polar  and 

equatorial  radii  amounting  to  four  (4)  and  five  (5) 


The  Law  of  Gravitation          97 

seconds  of  arc  were  found ;  some  observers  finding  the 
polar  radius  the  longer,  others,  the  equatorial.  Some- 
what later,  Auwers  collected  and  discussed  all  the 
observations,  made  at  the  leading  observatories  of  the 
world,  Greenwich,  Washington,  Radcliffe,  and  others; 
something  like  30,000  observations  made  by  nearly  one 
hundred  different  astronomers.  From  this  immense 
mass  of  material  he  finally  concluded  that  a  polar  com- 
pression of  i/4,oooth,  or  an  excess  of  o".5  of  the 
equatorial  over  the  polar  diameter,  would  best  represent 
all  the  observations. 

The  transits  of  Venus  across  the  disc  of  the  sun  in 
1874  and  in  1882  furnished  splendid  opportunities  for 
determining  the  size  of  the  solar  disc.  An  extremely 
accurate  measuring  instrument,  the  heliometer,  had  been 
devised  and  elaborate  preparations  for  observing  the 
transits  were  made.  During  the  preparatory  work  of 
adjusting  the  instruments  and  determining  their  con- 
stants, the  German  astronomers  made  many  long  series 
of  measurements  of  the  solar  diameter.  Five  instru- 
ments were  used,  measurements  with  the  same  instru- 
ment being  made  in  various  localities  by  the  same 
observer,  and  at  the  same  station  by  various  observers. 
In  all  some  2692  separate  measures  of  the  sun's  diame- 
ter were  made  by  twenty-six  (26)  observers.  This 
mass  of  data  was  thoroughly  discussed  by  Auwers  *  in 

*  Astronomische  Nachrichten,  vol.  cxxviii,  No.  3068,  December, 
1891. 

7 


98     Gravitation  versus  Relativity 

1891,  with  the  conclusion  that  the  polar  diameter 
exceeds  the  equatorial  by  the  very  minute  amount  of 
0.038  seconds  of  arc.  This  result  is  contrary  to  what 
would  normally  be  expected.  The  sun  is  a  rotating 
gaseous  body,  and,  under  all  laws  of  physics  and  me- 
chanics, the  equatorial  diameter  should  be  the  longer 
by  at  least  0^.05.  Newcomb,*  however,  refers  to  this 
investigation  of  Auwers  as  completely  setting  at  rest 
all  questions  as  to  the  sun's  shape,  and  concludes  that 
there  can  be  no  non-symmetrical  distribution  of  matter 
in  the  sun  sufficient  to  cause  any  appreciable  variation 
in  the  motions  of  any  planet. 

A  result  almost  identical  with  that  of  Auwers,  was 
obtained  by  Schur  and  Ambronne  in  1905.  They  made 
a  series  of  measures  of  the  sun's  diameter  with  a  six- 
inch  heliometer  during  a  period  of  nearly  thirteen  years, 
from  1890  to  1902,  finding  that  the  polar  diameter 
exceeds  the  equatorial  by  0^.025.  This  result,  as  well 
as  that  of  Auwers,  represents  the  mean  or  average  of 
all  the  observations,  regardless  of  the  time  at  which 
they  were  made.  The  individual  results  varied  in  dif- 
ferent years ;  in  some  years  the  polar  diameter  appeared 
the  greater,  in  other  years,  the  equatorial. 

This  question  as  to  a  possible  variability  in  the  shape 
of  the  sun  has  also  been  made  the  subject  of  many  in- 
vestigations. It  is  a  well-known  fact,  easily  confirmed 
by  any  amateur  observer,  that  the  visible  "spots"  on 

*  Astronomical  Constants,  Washington,  1895. 


The  Law  of  Gravitation          99 

the  sun's  surface  are  periodic  in  character ;  their  aver- 
age number  waxes  and  wanes  in  regular  periods  of 
eleven  years.  During  a  minimum,  practically  no  spots 
are  visible,  days  and  weeks  often  passing  without  a 
single  spot  marring  the  brilliant  solar  surface.  Then  a 
few  small  spots  appear,  and  gradually  the  number  and 
size  of  the  spots  increase,  until  after  the  lapse  of  five 
and  a  half  years  portions  of  the  surface  are  constantly 
covered  with  large  and  small  spots.  Occasionally 
spots  are  so  numerous  at  times  of  maxima,  as  to  form 
two  great  belts  around  the  sun,  one  on  each  side  of  the 
equator.  Now,  if  this  eleven-year  cycle  is  a  natural 
period  of  the  sun,  due  to  its  physical  condition  as  a 
rotating,  cooling  mass  of  gas,  then  it  is  highly  proba- 
ble that  it  expands  and  contracts  as  the  number  of 
spots  wax  and  wane ;  and  this  expansion  and  contraction 
would  be  reflected  in  the  measured  values  of  the  diame- 
ter. Practically  every  series  of  measures,  heretofore 
made,  show  periodic  variations  in  the  size  and  shape 
of  the  sun;  but  it  has  not  been  proved  that  such  varia- 
tions agree  in  period  with  the  sun-spots. 

In  his  investigation  of  the  long  series  of  observations, 
heretofore  referred  to,  Auwers  first  found  that  the 
measured  diameter  of  the  sun  varied  with  the  number 
of  visible  sun-spots.  He  afterwards  revised  his  discus- 
sion and  assigned  variable  "personal  equations"  to  the 
different  observers;  making  those  of  some  observers 
vary  with  the  time,  and  those  of  other  show  abrupt 


ioo    Gravitation  versus  Relativity 

changes.  With  such  set  of  variable  personal  equations, 
Auwers  was  enabled  to  reduce  the  apparent  variation  in 
the  observations,  so  that  all  semblance  of  periodicity 
disappeared.  In  other  words,  it  appeared  more  plausible 
to  Auwers  that  a  dozen  or  more  observers  should  each 
change  his  personal  habit  of  observation  by  just  the 
right  amount  and  at  exactly  the  right  time,  rather  than 
have  a  real  change  in  the  object  they  all  were  measur- 
ing. This  may  be  the  true  explanation  of  the  variations 
in  the  measured  diameter  of  the  sun;  but  it  is  evident 
that  it  is  perfectly  possible  to  get  any  pre-determined 
result  in  this  way. 

On  the  other  hand,  Ambronne  found  a  small  periodic 
variation  of  about  o".i  from  his  own  lorig  series  of 
measures,  but  this  variation  did  not  appear  to  bear  any 
relation  to  the  sun-spot  period.  A  revision*  of  this 
work  of  Ambronne,  together  with  an  investigation  of 
the  transit  of  Venus  observations,  confirms  a  variation 
of  about  the  size  found  by  Ambronne,  but  differs  from 
his  conclusion,  in  that  it  indicates  a  close  connection 
between  the  period  of  this  fluctuation  and  that  of  the 
sun-spots.  Some  faint  indications  of  short  period 
fluctuations  were  also  found ;  periods  varying  from  ten 
Ijours  to  twenty-eight  days. 

Thus  all  the  measures  made  of  the  visible  surface  of 


*An  Investigation  of  the  Figure  of  the  Sun  and  of  Possible 
Variations  in  its  Size  and  Shape,  by  Charles  Lane  Poor:  New 
York  Academy  of  Sciences,  vol.  xviii,  No.  9. 


The  Law  of  Gravitation,; ;/;  101 

the  sun  show  that  it  is  almost  exactly  a  spherical 
body,  but  indicate  that  its  shape  is  subject  to  minute 
fluctuations.  The  average  difference  of  the  diameters 
of  the  sun  cannot  be  more  than  a  small  fraction  of 
a  second  of  arc,  and  the  amplitude  of  the  fluctuation 
is  also  extremely  minute,  being  probably  not  over 
a  tenth  of  a  second  of  arc.  Such  minute  departures  from 
sphericity  are  far  too  small  to  have  any  appreciable 
effect  upon  the  larger  motions  of  the  planets,  and  the 
fundamental  approximation  of  Newton  and  his  suc- 
cessors in  regarding  the  sun  as  a  sphere  is  thus 
justified  for  the  relatively  crude  observations  of 
those  times.  But  the  sun,  clearly,  is  not  a  sphere 
of  uniform  density,  and  the  departure  from  sphericity, 
minute  though  it  may  be,  must  be  taken  into  account 
when  considering  the  intricate  motions  of  the  inner 
planets.  Any  deductions,  obtained  through  the  omis- 
sion of  this  non-sphericity,  are  clearly  approxima- 
tions ;  extremely  close  approximations,  but  approxima- 
tions nevertheless. 

The     SECOND     FUNDAMENTAL     APPROXIMATION     in 

all  planetary  theories  is  that  space  is  empty;  that, 
so  far  as  the  equations  of  motion  are  concerned, 
no  bodies  exist  other  than  the  sun,  the  planets,  and 
their  satellites.  In  every  mathematical  discussion  of 
the  motions  of  the  planets,  in  the  works  of  La  Place, 
Leverrier,  and  of  Newcomb,  the  solar  system  is  con- 
sidered as  formed  of  a  limited  number  of  material 


102  \Gr^v:itaik)xi  v.er$us  Relativity 

particles  of  certain  masses,  each  of  which  attracts  every 
other  one  according  to  Newton's  law  of  gravitation. 
The  first  of  these  material  particles  represents  the  sun, 
the  second,  Mercury,  the  third,  Venus,  the  fourth,  the 
earth-moon  system,  and  so  on  to  Neptune  and  his  satel- 
lite. The  planetoids,  the  comets,  the  matter  forming  the 
solar  corona,  and  any  and  all  other  matter,  which  may 
lie  about,  or  between  the  planets;  all  these  are  con- 
sidered as  absolutely  negligible.  All  the  tables  of  plane- 
tary motion,  all  the  formulas  and  deductions  of  celestial 
mechanics  are  based  upon  the  assumption  that  these 
smaller  bodies,  and  this  scattered  matter,  have  and  can 
have  no  effect  upon  the  motions  of  the  planets.  In  a 
first  approximation  to  the  planetary  motions,  this  as- 
sumption is  essential;  the  equations  of  celestial  me- 
chanics would  be  unworkable  if  an  attempt  were  made 
to  introduce  factors  representing  these  bodies.  As  will 
be  seen  in  a  subsequent  chapter,  the  mathematical  dif- 
ficulties in  tracing  the  paths  of  even  a  limited  number  of 
bodies,  are  almost  unsurmountable ;  to  attempt  to 
formulate  theories  of  motion  of  a  great  number  of 
heterogeneous  bodies  of  all  sizes  and  conditions  would 
be  hopeless. 

A  comparison  of  the  calculated  motions  of  the 
planets  with  their  observed  paths  shows  that  this  ap- 
proximation is  justified.  The  larger  motions  of  the 
planets  are  exactly  represented ;  the  more  minute  varia- 
tions in  motion  are  represented  with  extreme  accuracy. 


The  Law  of  Gravitation         103 

This  assumption  of  empty  space  serves,  thus,  not  only 
as  a  first,  but  as  a  second  approximation.  But  it  is  an 
approximation,  and  these  neglected  bodies  must  have 
some  effect  upon  the  motions  of  their  larger  fellows, 
and  such  effect  must,  in  the  course  of  time,  appreciably 
alter  the  motions  of  one  or  more  of  the  planets. 

Now  consider  for  a  moment  what  these  neglected 
bodies  really  are,  so  as  to  have  some  idea  as  to  the  order 
of  this  fundamental  approximation.  Comets  may  be 
dismissed  at  once  from  consideration;  they  are  erratic 
wanderers,  and  any  effect  that  they  might  have  upon 
the  motions  of  the  planets,  therefore,  would  be  momen- 
tary. There  is  not  the  slightest  indication  that  a  comet 
has  ever  affected,  in  any  measurable  degree,  the  motion 
of  even  the  smallest  planet  or  satellite. 

The  sun,  however,  is  surrounded  by  an  envelope  of 
matter,  an  envelope  which  becomes  visible  at  times  of 
eclipse.  This  brilliant  halo,  the  Corona,  has  been  known 
from  the  times  of  remotest  antiquity  as  one  of  the 
most  beautiful  of  all  natural  phenomena.  Since  the 
days  of  exact  science  it  has  been  drawn  many  times,  it 
has  been  photographed  times  without  number.  In  shape 
and  brilliancy  it  is  extremely  variable,  although  certain 
general  characteristics  are  maintained  from  eclipse  to 
eclipse.  The  inner  portion  appears  as  an  almost  con- 
tinuous ring  of  brilliantly  glowing  matter,  while  the 
outer  portions  are  made  up  of  radiating  bands  and 
filaments,  which  vary  greatly  in  length.  The  longest 


104    Gravitation  versus  Relativity 

streamers  are  usually  found  in  connection  with  those 
portions  of  the  solar  surface  to  which  the  sun-spots  are 
confined,  and  some  indications  point  to  a  real  connection 
between  the  shape  and  size  of  the  corona  and  the  sun- 
spot  period.  At  times  the  coronal  bands  and  streamers 
have  been  traced  to  great  distances  from  the  disc  of  the 
sun, — ten  or  more  diameters.  In  a  very  general  way  it 
may  be  said  that  the  corona  is  lens-shaped, — an  ellipsoid 
of  matter  with  its  longer  axis  in,  or  near,  the  plane  of 
the  sun's  equator. 

The  total  light  of  the  corona  is  two  or  three  times 
that  of  full  moon,  and  such  light  is  found  to  be  made 
up,  partly  of  reflected  sun-light,  partly  of  light  from 
a  self-luminous  gas.  Thus,  the  corona  must  consist  of 
a  great  mass  of  incandescent  gas,  intermixed  with  small 
particles  of  solid  or  liquid  matter.  Very  little,  how- 
ever, is  known  as  to  its  physical  constitution:  it  is 
generally  assumed  to  be  of  inconceivable  rarity,  its 
density  far  less  than  the  best  vacuum  produced  in  our 
laboratories.  This  is  purely  a  matter  of  speculation; 
nothing  definite  is  known  as  to  its  actual  density,  be- 
yond the  mere  fact  that  it  is  small  as  compared  to  the 
earth's  atmosphere.  The  conditions  in  the  immediate 
vicinity  of  the  sun  are  so  radically  different  from  any- 
thing known  on  the  earth  that  all  analogies  fail; — the 
temperature  at  the  solar  surface  is  some  8,000°  or  some 
10,000°,  and  the  force  of  gravity  there  is  some  27  times 
that  on  the  earth.  Laws  of  pressure  and  density  found 


The  Law  of  Gravitation         105 

under  laboratory  conditions  may  be  greatly  modified 
under  conditions  so  dissimilar.  At  these  temperatures 
elements  are  dissociated,  and  radiation,  or  light,  pres- 
sure is  a  distinct  factor :  planetary  gravitation  may  be 
at  play,  and  the  discrete  particles  of  the  corona  may 
travel  about  the  sun  in  independent  orbits.  It  is  clearly 
evident  that  the  corona  is  not  a  true  atmosphere, 
analogous  to  the  atmosphere  of  the  earth. 

But  the  solar  envelope  does  not  end  with  the  visible 
corona,  it  extends  far  out  beyond  the  orbit  of  the  earth. 
And  this  outer  portion  of  the  envelope,  unlike  the 
corona,  can  be  seen  by  any  one  on  any  clear,  moon-less 
evening  of  early  spring.  It  is  known  as  the  Zodiacal 
Light  and  it  appears  as  a  faint,  soft,  lens-shaped  beam 
of  light,  extending  up  from  the  horizon  along  the  path 
of  the  sun.  It  is  about  as  bright  as  the  Milky  Way, 
but  it  is  not  clearly  defined,  it  fades  away  gradually 
into  the  general  illumination  of  the  sky. 

The  spectroscope  shows  that  the  light  is  reflected 
sunlight.  It  is  reflected  from  bodies  too  minute  to  be 
seen  individually,  but  sufficiently  numerous  to  give  the 
faint  glow  of  the  Zodiacal  Light.  These  bodies,  what- 
ever their  origin,  form  an  immense  group  extending 
from  the  sun  to  far  beyond  the  orbit  of  the  earth; 
being  very  numerous  in  and  near  the  plane  of  the 
ecliptic,  densest  and  thickest  near  the  sun,  becoming 
fewer  and  more  scattered  as  the  distance  from  that 
body  and  from  the  ecliptic  increases.  These  bodies  are 


io6    Gravitation  versus  Relativity 

apparently  independent  planetesimals,  varying  in  size 
from  the  minutest  dust  particles  to  rocks  and  masses  of 
iron  of  appreciable  diameter;  each  revolving  about  the 
sun  in  its  own  independent  orbit.  Many  of  these  paths 
intersect  that  of  the  earth,  and  collisions  are  frequent. 
When  thus  entrapped  in  the  earth's  atmosphere,  these 
bodies  become  individually  visible  as  shooting-stars  and 
meteors.  Most  of  the  smaller  particles  are  completely 
fused  by  the  friction  of  the  atmosphere,  but  some  of 
the  larger  bodies  reach  the  surface  of  the  earth  in  the 
form  of  the  well  known  meteorites. 

These  bodies  enter  the  earth's  atmosphere  from  every 
direction  and  with  greatly  varying  velocities.  While 
comparatively  few  reach  the  surface  of  the  earth,  the 
number  of  those  consumed  in  the  upper  atmosphere  is 
enormous.  On  any  clear  night  half  a  dozen  may  be 
seen  in  the  course  of  a  few  minutes,  never  an  hour  will 
elapse  without  one  or  two  being  visible.  On  some 
nights,  at  certain  seasons  of  the  year,  the  number  that 
can  be  seen  from  a  single  spot  can  be  counted  by  the 
hundreds.  From  many  careful  counts,  made  at  dif- 
ferent seasons  and  at  widely  different  places,  it  has  been 
estimated  that  from  ten  to  twenty  million  meteors  enter 
the  earth's  atmosphere  each  day.  But  the  individual 
bodies  are  extremely  small :  from  photometric  observa- 
tions, Newcomb  showed  that  the  majority  of  the 
meteors  seen  weigh  less  than  a  single  grain.  The  largest 
and  brightest  observed  by  him  in  his  series  of  deter- 


The  Law  of  Gravitation         107 

minations,  did  not  weigh  more  than  a  quarter  of  an 
ounce. 

These  minute  bodies  are  consumed  in  the  upper 
atmosphere,  at  heights  ranging  from  40  to  100  miles 
above  the  surface,  and  the  products  of  their  combustion 
gradually  fall  to  the  earth's  surface  in  the  form  of  very 
finely  pulverized  dust.  Thus,  the  earth  is  gradually 
growing  larger,  but  at  an  extremely  slow  rate.  Even 
with  the  enormous  number  of  meteors,  as  above  esti- 
mated, the  growth  of  the  earth  can  hardly  exceed  a 
few  tons  per  day,  and  it  would  take,  at  the  present  rate, 
many  millions  of  years  to  increase  the  diameter  of  the 
earth  by  one  inch. 

Occasionally,  three  or  four  times  a  year,  meteors  are 
seen  to  fall  to  the  surface  of  the  earth,  and  many  have 
been  found  and  are  now  in  museums  and  cabinets. 
One  of  the  best  collections  of  such  meteorites  may  be 
seen  at  the  American  Museum  of  Natural  History  in 
New  York  City,  which  embodies  many  of  the  largest 
in  the  world.  Here  can  be  seen  and  handled  the  actual 
matter  from  outside  space. 

Space  is  not  empty :  and  any  deductions,  any  motions 
of  the  planets,  obtained  through  the  failure  to  take  into 
account  the  mass  of  minute  particles  traversing  space, 
are  approximations. 


CHAPTER  IV 

THE  MOTIONS  OF  THE  PLANETS 

THE  discussion  of  the  motions  of  the  planets  about 
the  sun  may  be  greatly  simplified  by  the  use  of  the  two 
fundamental  approximations,  discussed  in  the  last 
chapter.  The  individual  dimensions  of  the  various 
bodies  may  be  disregarded  in  the  first  instance,  and 
the  solar  system  may  be  considered,  mathematically,  as 
consisting  of  a  small  number  of  material  particles,  acted 
upon  by  their  mutual  attractions  only,  and  situated  and 
having  their  motions  in  empty  space.  In  discussing  the 
motions  of  such  a  system  of  imaginary  mathematical 
bodies,  Newton  assumed,  in  addition  to  the  law  of 
gravitation,  three  fundamental  laws  of  motion — three 
postulates,  to  use  the  word  made  prominent  by  the  Rela- 
tivity Theory.  These  laws  of  Newton,  however,  are 
simple  and  are  based  upon  the  experience  of  centuries. 
They  are : 

a.  Every  body  continues  in  its  state  of  rest,  or  of 
uniform  motion  in  a  straight  line,  unless  it  is 
compelled  to  change  that  state  by  forces  im- 
pressed upon  it. 

10$ 


The  Motions  of  the  Planets      109 

b.  The  change  in  motion  is  proportional  to  the 
force  impressed  and  takes  place  in  the  direc- 
tion of  the  straight  line  in  which  the  force  acts. 

c.  To  every  action  there  is  an  equal  and  opposite 
reaction ;  or,  the  mutual  actions  of  two  bodies 
are  always  equal  and  oppositely  directed. 

The  principles  involved  in  these  laws  were  known  to 
Galileo,  but  to  Newton  is  due  the  clear  enunciation  of 
them.  The  first  law  merely  means  that  in  empty  space 
uniform  rectilinear  motion  is  just  as  natural  as  rest, 
that  such  motion  implies  no  physical  cause  and  requires 
no  explanation.  Change  in  motion,  however,  is  dif- 
ferent; change  of  any  kind,  in  speed  or  in  direction, 
implies  force,  external  force,  acting,  or  impressed, 
upon  the  body.  This  idea  of  motion  and  force  is 
radically  different  from  the  conception  of  Aristotle  and 
the  older  philosophers,  who  thought  that  whenever  a 
body  was  in  motion,  some  force  must  operate  to  keep  it 
moving.  On  the  contrary,  no  force  whatever  is  re- 
quired to  keep  a  body  in  motion ;  force  is  required  only 
to  change  the  motion.  If,  through  the  action  of  some 
force,  a  body  be  set  in  motion  and  the  force  ceases  then 
to  act,  the  body  will  not  stop  when  the  force  ceases, 
but  will  continue  forever  to  move  forward  in  a  straight 
line  at  a  uniform,  or  constant,  speed. 

Now,  as  change  in  motion,  either  in  speed  or  in  direc- 
tion, implies  the  action  of  a  force,  so  the  amount  of 
this  change  may  be  used  as  a  measure  of  the  size  of  the 


i  io    Gravitation  versus  Relativity 

force.  If  a  certain  force  act  upon  a  body,  initially  at 
rest,  for  one  second,  so  that  at  the  end  of  the  second 
the  body  is  moving  with  a  speed  of  ten  (io)  feet  per 
second;  and  if  a  second  force  act  upon  a  similar  body 
and  impart  to  it  a  speed  of  twenty  (20)  feet  per 
second;  then  is  the  second  force  twice  as  great  as  the 
first.  The  change  in  motion  is  proportional  to  the  force 
impressed.  It  will  be  noted  that  the  element  of  time  also 
enters,  for  the  longer  a  force  acts  upon  a  body,  the 
greater  its  effect  and  the  swifter  will  be  the  resultant 
motion  of  the  body.  All  of  this  is  shown  in  Figure  16, 
where  A  and  B  represent  two  cannon  of  different 
lengths  and  taking  different  powder  charges,  but  firing 


B  A 

Fig.  16.    Forces  and  Trajectories. 

shells  of  exactly  the  same  weight  and  size.  The  heavier 
powder  charge  in  B  produces  a  muzzle  velocity,  at  the 
end  of  a  definite  fraction  of  a  second,  of  exactly  twice 


The  Motions  of  the  Planets      m 

that  of  A ;  hence  the  explosive  force  of  B  is  twice  that 
of  A.  On  leaving  the  muzzles  of  the  guns,  the  force  of 
the  explosion  ceases  to  act  upon  the  shells  and,  were  it 
not  for  the  friction  of  the  air  and  the  attraction  of  the 
earth,  the  shells  would  travel  forever  in  straight  paths 
and  at  constant  speeds.  But,  no  sooner  have  the  pro- 
jectiles left  the  muzzles  than  they  are  subject  to  two 
forces,  one  the  resistance  of  the  atmosphere  and  the 
other  the  attraction  of  the  earth.  The  former  acts 
directly  in  the  line  of  motion  and  slows  up  the  speed 
of  the  shells,  but  does  not  change  the  direction  of 
motion;  the  attraction  of  the  earth,  however,  pulls 
the  shells  downward  and  causes  them  to  describe  curved 
paths,  finally  bringing  both  to  the  ground.  If  the  gun  A 
were  greatly  lengthened  so  that  the  explosive  force  of 
the  powder  had  a  longer  time  interval  to  act  upon  the 
shell,  then  would  its  velocity  be  increased  and  the 
smaller  force  might  thus  impart  the  same  velocity  as 
the  greater  force  in  B. 

Now  suppose  the  air  removed,  so  that  there  will  be 
no  friction,  and  the  gun  mounted  on  the  top  of  a  tower, 
so  as  to  shoot  projectiles  horizontally  at  various  speeds. 
Upon  leaving  the  muzzle  of  the  gun,  the  shell  will  be 
acted  upon  solely  by  the  attraction  of  the  earth.  If  the 
shell  merely  roll  out  of  the  muzzle,  it  will  fall  in  a 
straight  line  to  the  surface  of  the  earth;  if  the  charge 
be  sufficient  to  give  the  shell  a  small  muzzle  velocity, 
it  will  describe  a  curved  path  and  fall  to  the  surface 


Gravitation  versus  Relativity 

some  miles  from  the  foot  of  the  tower.  As  the  charge 
is  increased,  the  muzzle  velocity  becomes  greater  and 
the  range  of  the  projectile  longer.  In  every  case,  the 
explosive  force  of  the  charge  ceases  the  instant  the 
shell  leaves  the  gun;  from  that  moment,  the  only  force 
acting  upon  the  shell  is  the  attraction  of  the  earth,  and 
it  is  this  force  of  gravity  that  pulls  the  shell  out  of  its 
straight  line  path  and  causes  it  to  travel  in  the  curved 
trajectory. 


Fig.  17.    Projectiles  near  the  Earth. 

The  ordinary  muzzle  velocity  of  a  shell  is  something 
less  than  half  a  mile  per  second  and  such  a  shell  falls 
to  the  earth  a  few  miles  from  the  foot  of  the  tower. 
If  this  velocity  could  be  increased  to  something  over 
four  miles  per  second,  the  path  of  the  bullet  would  be 


The  Motions  of  the  Planets      113 

nearly  four  thousand  miles  long,  the  bullet  coming  to 
the  earth  at  this  distance  from  the  tower.  If  the 
muzzle  velocity  were  increased  beyond  this  point  to 
about  4.9  miles  per  second,  then  the  shell  would  miss  the 
earth  entirely  and  would  travel  about  the  earth  in  a 
circle.  As  the  circumference  of  the  earth  is  approxi- 
mately 25,000  miles,  it  would  take  such  a  projectile 
travelling  4.9  miles  each  second  about  5100  seconds,  or 
i  hour  24.7  minutes,  to  complete  one  circuit  about  the 
earth  and  to  return  to  the  starting  point. 

If  the  muzzle  velocity  were  still  further  increased, 
the  path  of  the  projectile  would  become  an  ellipse  and 
its  period  would  become  longer.  When  the  velocity 
reached  6.94  miles  per  second,  the  path  would  become 
a  parabola  and  the  bullet  would  never  return  to  the 
starting  point. 

These  illustrations  may  help  one  to  understand  the 
motions  of  the  planets.  The  force  of  gravitation,  the 
attraction  of  the  sun,  keeps  the  planets  in  their  respec- 
tive orbits,  but  it  does  not  explain  or  account  for  the 
original  motion,  for  the  muzzle  velocity  of  the  shell,  so 
to  speak.  Once  the  planets  were  set  in  motion  about 
the  sun,  gravitation  took  charge  and  regulated  and 
controlled  the  various  paths,  but,  somehow,  in  some 
inexplicable  way  the  system  was  set  in  motion  and  each 
planet  given  an  initial  velocity.  There  have  been  many 
attempts  to  explain  this  initial  velocity,  to  show  how  it 
may  have  been  caused,  but  such  attempts  are  specula- 


H4    Gravitation  versus  Relativity 

tions,  purely  and  simply.  The  law  of  gravitation  deals 
with  the  facts  of  present  day  motions,  it  does  not  deal 
with  the  primal  cause  of  motions. 

Now  consider  a  very  simple,  ideal  case, — that  of  two 
perfect  spheres  of  uniform  density  and  separated  by  a 
measurable,  finite  distance.  It  has  been  seen  that  for 
spherical  bodies  the  force  of  attraction  varies  only 
with  the  distance  between  their  centres,  and  that,  if  the 
masses  of  the  bodies  be  respectively  M  and  m,  and  the 
distance  between  the  centres  r,  then  the  mutual  force 
of  attraction  is  proportional  to : 


Mm 


If  these  were  the  only  bodies  in  the  universe  and  if  they 
were  at  absolute  rest  at  the  beginning  of  time,  then  this 
force  of  attraction  would  draw  them  towards  each 
other,  with  different  and  rapidly  increasing  speeds. 
As  the  same  force  acts  upon  both  bodies,  the  smaller 
will  move  the  faster,  and  the  bodies  will  finally  come 
together  at  a  point  nearer  the  initial  position  of  the 
larger.  The  point  at  which  the  centres  would  meet 
(disregarding  the  actual  dimensions  of  the  bodies)  is 
what  is  commonly  known  as  the  centre  of  gravity  of 
the  system. 

If,  however,  at  the  initial  moment,  the  two  bodies 
were  in  motion  relative  to  each  other  and  in  any  direc- 


The  Motions  of  the  Planets      115 

tion  inclined  to  the  line  joining  their  respective  centres, 
then,  instead  of  falling  directly  toward  the  common 
centre  of  gravity,  each  ball  would  describe  around  such 
centre  of  gravity  one  of  several  simple  geometrical 
curves.  The  centre  of  gravity  would  remain  at  rest, 
or  would  partake  of  any  uniform  rectilinear  motion  of 
the  two  bodies.  The  type  and  size  of  the  curves  de- 
scribed by  the  bodies  depend  solely  upon  the  relative 
motion  of  the  two  bodies  at  the  initial  instant  of  time, 
just  as  the  range  of  the  gun  depends  upon  the  muzzle 
velocity  of  the  projectile.  These  curves  may  be  any  of 
the  curves  known  in  geometry  as  conic  sections,  but  in 
the  vast  majority  of  cases  actually  occurring  in  nature 
the  curve  is  that  known  as  an  ellipse.  If  both  bodies 
are  of  the  same  size,  then  the  respective  elliptical  paths 
are  also  of  the  same  size,  but,  if  the  balls  differ  in  size, 
then  the  two  orbits  will  also  differ,  the  smaller  body, 
however,  will  travel  in  the  larger  orbit.  If  the  two 
bodies  differ  tremendously  in  size,  as  do  the  sun  and 
the  planets,  then  the  common  centre  of  gravity  will  be 
very  near  the  centre  of  the  larger  body,  may  indeed  lie 
within  that  body,  and  the  smaller  body  will  apparently 
describe  an  ellipse  about  the  larger.  In  the  case  of  the 
earth  and  the  sun,  the  latter  body  is  some  330,000 
times  as  large  as  the  earth,  and,  therefore,  the  radius 
of  the  sun's  orbit  is  only  some  280  miles  as  against  the 
93,000,000  miles  of  the  earth's. 

In  the  relatively  crude  observations  at  the  time  of 


n6    Gravitation  versus  Relativity 

Kepler  it  could  make  little  difference  whether  the  earth 
was  considered  as  traveling  about  the  centre  of  the  sun, 
or  about  the  common  centre  of  gravity  of  the  two 
bodies.  Not  so,  however,  with  the  accuracy  attainable 
with  modern  instruments.  But  it  is  a  very  simple 
proposition  to  find  and  to  use  the  orbit  of  the  earth 
relative  to  the  sun's  centre  in  place  of  the  actual  orbit 
about  the  centre  of  gravity.  This  relative  orbit  will  be 
exactly  similar  to  the  actual  one  about  the  centre  of 
gravity,  except  that  it  will  be  larger,  will  be  magnified 
in  the  proportion  of  the  total  combined  mass  of  the  two 
bodies  to  that  of  the  sun  alone.  That  is,  the  relative 
orbit  of  the  earth  about  the  sun  is  to  the  actual  orbit 
about  the  centre  of  gravity  as  M  +  m  is  to  M,  or,  in 
figures,  as  330,000  +  i  is  to  330,000.  The  relative 
orbit  of  the  earth  is  thus  about  280  miles  larger  than 
the  actual  one  about  the  centre  of  gravity.  And  it  is 
this  relative  orbit  that  is  really  used  in  all  astronomical 
work,  for  the  sun's  centre  is  a  distinct  point,  which  can 
be  accurately  located  in  the  heavens,  while  the  position 
of  the  centre  of  gravity  can  only  be  determined  by 
calculation.  Further,  the  centre  of  the  sun  is  a  common 
point  to  which  can  be  referred  the  motions  of  all  the 
planets,  instead  of  using  for  each  planet  the  differently 
located  centre  of  gravity  of  the  system,  sun  and  earth, 
sun  and  Jupiter,  and  so  forth. 

Thus,  under  the  ideal  conditions  imposed,  namely, 
that  there  are  but  two  bodies  in  the  universe  and  that 


The  Motions  of  the  Planets      117 

both  of  these  bodies  are  spheres  of  uniform  density, 
the  path  of  the  smaller  body  about  the  centre  of  the 
larger  will  be  a  conic  section — an  ellipse,  a  parabola,  or 
an  hyperbola.  The  character  of  the  curved  path  and 
its  size  depend  solely  upon  the  initial  velocity  of  the 
smaller  body  and  its  distance  from  the  larger :  the  exact 
shape  of  the  path  and  its  position  in  space  depend,  not 
alone  upon  the  velocity  of  the  smaller,  but  also  upon 
the  direction  of  its  motion.  This  is  shown  in  Figure 
1 8,  where  planets  pass  through  P  in  various  directions, 


Fig.  18.    Ellipses  of  the  Same  Size. 

but  always  with  the  same  speed.  In  every  case  the 
orbit  is  elliptical,  the  sun  being  at  one  focus,  and  the 
major  axes  of  all  the  ellipses  are  all  of  the  same  length : 


n8    Gravitation  versus  Relativity 

AA'  is  equal  to  BB'  and  to  CC :  but  the  ellipticity  of 
the  various  orbits  and  the  positions  of  their  respective 
perihelia  are  determined  by  the  directions  in  which  the 
planets  pass  through  P. 

If,  therefore,  the  velocity  of  the  planet  at  the  point, 
P,  is  known,  together  with  the  direction  in  which  it  is 
moving,  the  complete  path  that  it  will  describe  can  be 
found ;  and,  when  once  the  path  is  known,  the  position 
of  the  body  at  any  time  can  be  readily  calculated.  This 
is  the  celebrated  Problem  of  Two  Bodies,  which  was 
completely  solved  by  Newton.  The  simple  methods  of 
ordinary  mathematics,  however,  are  not  sufficient  for 
this  purpose,  and  Newton  was  obliged  to  devise  special 
mathematical  methods  and  formulas,  which  have  since 
been  developed  into  that  powerful  branch  of  mathe- 
matical analysis,  known  as  Calculus.  These  methods  are 
too  intricate  to  be  explained  here,  but  they  are  well 
known  to  all  mathematicians,  and  can  be  found  in  any 
ordinary  text-book  of  mathematics. 

The  Problem  of  Two  Bodies  may  be  stated  thus: 

Given,  at  any  moment  of  time,  the  positions,  musses, 
and  velocities  of  two  spherical  bodies,  acted  upon  solely 
by  their  mutual  attractions;  required  their  positions  and 
motions  at  any  future  time. 

This  is  the  problem  that  Newton  solved.  He  showed 
that  the  two  bodies  would  describe  forever  certain  par- 
ticular paths,  or  orbits  as  the  astronomers  call  them; 
the  exact  shape  of  the  path  depending  in  each  case  upon 


The  Motions  of  the  Planets      119 

the  motions  of  the  bodies  at  the  moment  of  starting. 
When  the  two  bodies  of  the  problem  become,  one  the 
sun  and  the  other  any  one  of  the  planets,  then  Newton 
showed  that  the  particular  path  of  the  planet  would  be 
an  ellipse,  as  described  by  Kepler,  and  that  the  planet 
would  move  in  such  ellipse  at  the  speed  called  for  by 
Kepler's  second  law.  Thus,  in  other  words,  Kepler's 
two  laws  of  planetary  motion  are  the  direct  and  neces- 
sary result  of  the  law  of  gravitation,  and  these  two  laws 
of  motion  are  sufficient  to  predict  the  position  of  the 
planet  at  any  future  time. 

Now  in  classifying  the  paths  of  the  various  planets 
and  in  locating  their  positions  in  space,  astronomers 
make  use  of  certain  conventional  terms,  which  are  very 
simple  and  which  ought  to  be  fully  understood.  They 
call  the  quantities,  which  define  the  size,  shape,  and 
position  of  the  path  of  a  planet,  the  "elements  of  the 
orbit."  These  elements  are  six  in  number :  two  of  them 
determine  the  plane  in  which  the  orbit  lies;  three  of 
them,  the  shape,  size,  and  position  of  the  path  in  the 
plane  of  motion;  and  the  sixth,  the  position  of  the 
planet  in  the  curved  path.  The  first  two  are  the  *  'in- 
clination of  the  orbit"  and  the  "longitude  of  the  node," 
and  their  names  clearly  indicate  their  purely  geometric 
character.  In  all  astronomical  work,  the  plane  of  the 
earth's  orbit,  the  ecliptic,  is  taken  as  the  fundamental 
plane  of  reference;  and  the  inclination  of  an  orbit  is 
merely  tne  angle  which  the  orbit  makes  with  this  funda- 


120    Gravitation  versus  Relativity 

mental  plane  of  reference.  These  two  planes  intersect 
in  a  straight  line  passing  through  the  sun,  called  the 
"line  of  nodes."  In  one  part  of  this  line,  the  planet,  as 
it  travels  about  the  sun,  will  pass  from  the  south  to  the 
north  side  of  the  ecliptic,  and  this  point  is  called  the 
ascending  node.  The  direction  in  which  this  point  lies 
is  determined  by  its  longitude,  and  this  longitude  is  the 
element  called  "longitude  of  the  node." 

The  three  elements  which  determine  the  shape,  size, 
and  position  of  the  orbit  in  the  plane  of  motion  are  the 
"eccentricity,"  the  "major  axis,"  and  the  "longitude  of 
the  perihelion."  The  first  two  of  these  have  the  ordinary 
geometric  meaning,  while  the  third  has  been  fully 
defined  in  a  former  chapter.  It  defines  the  direction  in 
which  the  axis  of  the  orbit  lies  in  space.  The  five  ele- 
ments, so  far  defined,  are  clearly  shown  on  the  following 
diagram,  which  needs  no  explanation. 

These  six  elements,  which  determine  the  motion  of  a 
planet  about  the  sun,  are  tabulated  as : 

a  =  the  semi  major  axis  of  the  ellipse,  or  one-half 
the  greatest  diameter  of  the  curved  path. 

e  =  the  eccentricity  of  the  orbit:  a  fraction  from 
which  can  be  determined  the  exact  shape  of 
the  path.  This  becomes  zero  for  a  circular 
path,  and  unity  for  a  parabolic  orbit. 

n  =  the  longitude  of  the  perihelion:  an  angle  (or 
rather  the  sum  of  two  angles)  which  locates 
the  point  of  nearest  approach  of  the  planet 
to  the  sun. 


The  Motions  of  the  Planets      121 

=  the  longitude  of  the  node:  an  angle,  which 
gives  the  direction  of  the  point  in  which 
the  planet  passes  from  the  south  to  the  north 


Fig.  19.    The  Elements  of  a  Planet's  Orbit. 

side  of  the  ecliptic,  or   plane  of  the  earth's 

orbit. 
i  —  the  inclination  of  the  orbit:  the  angle  between 

the  plane  of  the  planet's  orbit  and  that  of  the 

earth. 
T  —  the  time  of  perihelion  passage:  the  exact  hour, 

minute,  and  second  at  which  the  planet  passes 

the  perihelion. 


The  sixth  element  is  required  to  determine  the  exact 
position  of  the  planet  in  its  path,  for  it  is  conceivable 
that  there  might  be  three  or  four  separate  bodies  all 


122    Gravitation  versus  Relativity 

traveling  about  the  sun  in  the  same  path,  one  behind 
the  other.  This  element  may  be  the  longitude  of  the 
planet  as  seen  from  the  sun  at  a  given  date,  January 
I,  1900,  for  example,  or  else  the  precise  hour,  minute, 
and  second  at  which  it  passes  through  a  definite  point 
of  the  orbit,  as  the  perihelion,  or  the  node. 

Now  so  long  as  there  are  but  two  bodies  in  the 
system,  these  six  elements  are  constant,  and  the  smaller 
body  will  travel  for  ever  around  and  around  in  its  un- 
varying path.  From  these  elements  the  actual  position 
of  the  body  at  any  time,  past,  present,  or  future,  can  be 
calculated  by  very  simple  formulas. 

If,  however,  a  third  body  be  introduced  into  oui 
ideal  universe,  then  the  motions  of  the  bodies  are  no 
longer  simple  and  easily  calculated.  In  fact,  the  paths 
of  the  three  bodies  become  so  complicated  as  to  defy  any 
mathematical  description.  Newton  failed  to  find  a  so- 
lution of  this  problem;  and  every  mathematician  since 
his  time  has  likewise  failed.  Yet 
The  Problem  of  Three  Bodies  is  simple : 

Given,  at  any  moment  of  time,  the  positions,  masses, 
and  velocities  of  three  spherical  bodies;  find  their  mo- 
tions thereafter,  and  their  positions  at  some  definite 
future  instant  of  time. 

Acted  upon  solely  by  their  mutual  attractions,  the 
paths  that  these  bodies  will  describe  are  fixed,  the 
positions  that  they  will  occupy  at  some  future  date  are 
absolutely  predetermined.  Once  the  system  is  set  in 


The  Motions  of  the  Planets      123 

motion,  the  future  of  each  body  is  fixed,  its  path  pre- 
ordained. Thus  the  problem  has  a  definite  solution,  but, 
unfortunately,  no  mathematician  has  yet  been  able  to 
find  that  solution.  The  difficulty  is  entirely  with  the 
present  day  methods  of  mathematical  research  and  cal- 
culation; these  methods  are  inadequate.  The  beautiful 
method,  devised  by  Newton  to  solve  the  problem  of 
two  bodies,  fails  completely,  when  applied  to  a  system 
of  three  or  more  bodies.  Under  certain  special  con- 
ditions, mathematicians  have  been  able  to  find  an 
approximate  solution  of  the  problem,  but  even  such 
approximate  solution  is  extremely  intricate.  No  solu- 
tion of  the  general  problem  has  been  found. 

This  statement,  found  in  all  books,  may  need  a  little 
qualification,  for  it  is  possible,  in  very  many  cases,  to 
trace  out  the  paths  of  the  bodies,  step  by  step.  To  take 
a  concrete  illustration,  the  positions  and  velocities  of 
the  sun,  the  earth,  and  Jupiter  are  known  for  today. 
These  velocities  vary  with  the  time,  but  for  some  very 
short  interval  of  time  they  may  be  considered  as  con- 
stant. That  is,  for  a  single  day,  the  motions  of  each 
of  these  bodies  may  be  considered  as  being  uniform  and 
in  straight  lines.  The  amounts  and  directions  of  these 
motions  depend,  of  course,  upon  the  mutual  attractions 
of  the  three  bodies,  and  such  amounts  can  be  accurately 
calculated.  Thus,  from  the  measured,  or  known  posi- 
tions and  velocities  of  the  three  bodies  today,  it  is  a 
matter  of  direct  calculation  to  predict  where  they  will 


124    Gravitation  versus  Relativity 

be  tomorrow.  Then  from  the  positions  and  velocities 
of  tomorrow,  by  the  same  process,  can  be  calculated  the 
positions  for  the  next  day.  Thus  step  by  step,  it  is 
possible  to  trace  out  the  paths  of  the  three  bodies;  the 
difficulty  is  in  the  shortness  of  the  steps,  and  in  the  time 
required  to  make  the  calculations  for  each  step.  The 
calculations  are  long  and  it  would  take  considerably 
more  than  a  working  Jay  for  a  mathematician  to  trace 
out  the  path  of  the  earth  for  twenty- four  hours. 
Further,  in  many  cases,  the  speed  and  direction  of 
motion  of  a  body  change  so  rapidily,  that  a  day  even  is 
too  long  an  interval  between  steps.  In  tracing  the  mo- 
tion of  a  certain  comet,  it  became  necessary  to  shorten 
the  interval  to  fifteen  minutes;  to  take  ninety-six  steps 
to  trace  its  path  for  a  single  day.  It  required  one 
month's  time,  six  hours  a  day,  to  thus  trace  the  path 
of  this  comet  for  one  day.  By  still  further  reducing 
the  length  of  the  steps,  practically  every  problem  could 
be  solved ;  but  it  might  require  years,  or  even  centuries, 
to  make  the  calculations  necessary  to  trace  the  various 
paths  of  a  system  of  bodies  for  a  single  week.  The 
real  problem  is  that  of  finding  a  mathematical  short- 
cut; a  short-cut,  which  will  give  accurate  results  and 
save  this  immense  labor. 

The  special  conditions,  which  allow  of  approximate 
solutions,  are  those  found  in  the  solar  system — one 
great  dominating  central  body  accompanied  by  a 
number  of  relatively  very  small  and  distant  bodies.  In 


The  Motions  of  the  Planets      125 

this  case  the  central  body  primarily  controls  the  mo- 
tions of  its  companions  and  holds  them  to  approximate 
orbits.  The  interactions  of  the  smaller  bodies  merely 
cause  temporary  irregularities  in  these  orbital  motions. 
This  is  shown  in  the  following  diagram,  where  S, 
the  central  body,  is  very  large  in  comparison  with  the 
planets,  E  and  J. 


Fig.  20.    Motions  of  a  System  of  Bodies. 

If  the  earth  and  the  sun  were  the  sole  bodies  in  the 
universe,  then  the  earth  would  travel  about  the  sun  in 
an  elliptic  path,  as  shown.  But  Jupiter  attracts  both  the 
earth  and  the  sun,  and  attracts  these  bodies  differently. 
When  the  earth  is  directly  between  the  sun  and  Jupiter, 
as  at  E  in  the  diagram,  Jupiter  pulls  both  bodies  directly 
towards  itself,  but  as  the  earth  is  the  nearer,  the  pull 


126    Gravitation  versus  Relativity 

upon  the  earth  is  the  stronger.  The  respective  distances 
of  the  earth  and  the  sun  from  Jupiter  are  4.2  and  5.2, 
and  the  force  of  attraction  of  Jupiter  upon  these  two 
bodies  varies  inversely  as  the  squares  of  these  two 
numbers.  The  squares  are  respectively  17.6  and  27.0, 
which  numbers  are  very  closely  in  the  ratio  of  2  to  3. 
That  is,  the  attraction  of  Jupiter  on  the  earth  is  about 
il/2  times  its  attraction  on  the  sun,  as  is  indicated  by 
the  lengths  of  the  herring-bone  arrows  in  the  diagram. 
But  it  is  only  the  difference  of  these  attractions  that 
can  affect  the  motion  of  the  earth  about  the  sun,  and 
this  difference  amounts  to  only  a  small  fraction  of  the 
whole  attraction  of  Jupiter  upon  the  earth.  Further, 
the  sun  is  approximately  one  thousand  times  larger 
than  Jupiter  and  only  one  quarter  the  distance  from  the 
earth ;  so  that  the  actual  attraction  of  the  sun  upon  the 
earth  is  some  16,000  times  that  of  Jupiter.  The  differ- 
ential attraction  of  Jupiter  upon  the  earth,  therefore, 
amounts  to  only  about  one  fifty-thousandth  part  of  the 
direct  attraction  of  the  sun.  This  minute  effective  pull 
of  Jupiter  in  a  direction  contrary  to  that  of  the  sun 
will,  momentarily  and  to  a  very  slight  degree,  straighten 
out  the  path  of  the  earth:  the  curvature  of  the  earth's 
orbit  will  be  slightly  diminished,  and  the  orbit  itself 
made  a  trifle  larger. 

The  earth  moves  faster  in  its  orbit  than  does  Jupiter, 
and  a  few  months  later  the  three  bodies  will  be  in  the 
respective  positions,  S,  E',  and  J',  at  which  time  Jupiter 


The  Motions  of  the  Planets      127 

is  equally  distant  from  the  earth  and  the  sun.  The  at- 
tractive force  of  Jupiter  upon  the  two  bodies  is  now  the 
same  in  amount,  but  it  acts  in  different  directions. 
Again  only  a  small  part  of  the  actual  force  of  Jupiter 
upon  the  earth  is  effective  in  disturbing  the  motion 
about  the  sun,  and  this  disturbing  force  is  only  a  very 
minute  fraction  of  the  direct  attraction  of  the  sun.  In 
this  position  of  the  three  bodies,  the  effect  of  Jupiter 
is  to  draw  the  earth  and  sun  together,  to  make  the 
earth's  orbit  a  little  more  curved  and  slightly  smaller 
than  it  otherwise  would  be. 

As  Jupiter  and  the  earth  move  thus  around  the  sun 
in  their  respective  paths,  they  constantly  change  their 
relative  positions,  and,  with  the  change  in  position,  the 
disturbing  effect  of  Jupiter's  attraction  upon  the  motion 
of  the  earth  changes.  At  times  it  makes  the  orbit 
slightly  larger,  at  times  slightly  smaller.  But,  after 
the  lapse  of  some  399  days,  the  earth  will  again  be 
directly  between  the  sun  and  Jupiter;  not,  however,  at 
E,  but  at  E"  about  i/ioth  of  the  circumference  in  ad- 
vance of  E.  At  this  point  the  action  of  Jupiter  will  be 
similar  to  what  it  was  when  the  earth  was  at  E:  the 
curvature  of  the  earth's  orbit  being  diminished  and  the 
orbit  itself  made  a  trifle  larger.  If  the  orbits  of  both 
Jupiter  and  the  earth  were  circles,  this  effect  would  be 
identical  every  time  the  planets  came  into  line ;  it  would 
be  repeated  over  and  over  again  every  399  days.  But 
the  orbits  are  not  circular,  they  are  elliptical,  and  so 


128    Gravitation  versus  Relativity 

when  Jupiter  returns  to  opposition,  the  relative  dis- 
tances between  the  planets  and  the  sun  will  have 
changed,  and  the  perturbation  of  Jupiter  upon  the 
earth's  motion  will  not  be  the  same  as  before.  It  will 
be  the  same  in  character,  but  not  the  same  in  amount. 
Thus  the  perturbation  depends  upon  the  part  of  the 
orbit  in  which  the  opposition  happens;  it  will  change 
in  amount  as  the  opposition  falls  in  different  seasons  of 
the  year.  It  is  still  periodic,  it  goes  through  regularly 
recurring  changes,  but  the  period  is  not  the  simple  one 
of  399  days.  The  distance  of  the  earth  from  the  sun 
increases  and  diminishes  again  during  one  year,  the 
distance  of  Jupiter  from  the  sun  fluctuates  with  its 
periodic  time  of  nearly  twelve  years  (n.86  years), 
and  thus  the  actual  value  of  the  perturbation,  or  dis- 
turbance in  the  earth's  motion,  depends  upon  the  399- 
day  period,  the  365-day  period,  and  the  1 1.86-year 
period,  and  upon  other  modifying  periods  too  com- 
plicated to  mention. 

The  attraction  of  the  earth  has  a  reciprocal  effect 
upon  the  motion  of  Jupiter.  When  Jupiter  pulls  the 
earth  outward  and  enlarges  its  orbit,  the  earth  pulls 
Jupiter  inwards  and  makes  its  orbit  smaller;  when 
Jupiter  causes  the  earth  to  move  faster  in  its  path,  the 
earth  acts  as  a  brake  and  slows  down  the  speed  of 
Jupiter.  Thus  the  actual  motions  of  the  two  bodies 
about  the  sun  are  extremely  complicated;  they  move 
in  wavy,  snake-like  curves,  curves  that  are  compounded 


The  Motions  of  the  Planets      129 

of  all  sorts  of  motions  and  periods.  The  planets  do  not 
travel  in  elliptic  orbits  and  the  laws  of  Kepler  are  not 
true. 

From  the  time  of  Newton,  it  has  been  known  that 
Kepler's  laws  are  mere  approximations,  computer's 
fictions,  handy  mathematical  devices  for  finding  the  ap- 
proximate place  of  a  planet  in  the  heavens.  They  apply 
with  greater  accuracy  to  some  planets  than  to  others. 
Jupiter  and  Saturn  show  the  greatest  deviations  from 
strictly  elliptic  motion.  The  latter  body  is  often  nearly 
a  degree  away  from  the  place  it  would  have  been  had 
its  motion  about  the  sun  been  strictly  in  accord  with 
Kepler's  laws.  This  is  such  a  large  discrepancy,  that 
it  can  be  detected  by  the  unaided  eye.  The  moon  is 
approximately  half  a  degree  in  diameter,  so  that  the 
discrepancy  in  the  motion  of  Saturn  is  about  twice  the 
apparent  diameter  of  the  moon.  In  a  single  year, 
during  the  course  of  one  revolution  about  the  sun,  the 
earth  may  depart  from  the  theoretical  ellipse  by  an 
amount  sufficient  to  appreciably  change  the  apparent 
place  of  the  sun  in  the  heavens.  This  departure  from 
strictly  elliptic  motion  is  not  large  enough,  however,  to 
be  detected  without  the  aid  of  instruments,  but  it  is 
sufficiently  large  to  be  detected  by  the  ordinary  sextant 
of  the  navigator. 

The  real  problem  of  the  mathematical  astronomer  is 
to  find  some  method,  or  methods,  of  approximately 
representing  the  actual  paths  of  the  planets;  methods 


130    Gravitation  versus  Relativity 

by  which  the  approximations  may  be  made  closer  and 
closer,  as  the  astronomer  is  willing  to  spend  more  and 
more  time  on  his  calculations.  The  scheme  of  elliptic 
orbits  for  the  larger  planets,  of  parabolic  orbits  for 
comets,  furnishes  the  first  step  in  the  elaborate  system 
of  approximations  that  has  been  built  up.  The  second 
step  is  involved  in  extremely  complicated  mathematics, 
but  the  underlying  principle  is  not  difficult  to  under- 
stand. 

The  mathematician  considers  the  planet  as  always 
traveling  about  the  sun  in  an  elliptic  orbit,  but  considers 
the  orbit  itself  as  constantly  changing  in  size  and  shape. 
He  thinks  of  the  planet  as  a  bead  strung  on  a  flexible 
wire,  and  this  wire,  which  represents  the  orbit,  as  be- 
ing pushed  and  pulled  into  various  shapes  by  the  action 
of  the  other  planets;  the  bead  always  remains  on  the 
wire,  but  the  wire  is  distorted  into  ellipses  of  various 
sizes  and  shapes.  This  may  seem  a  rather  round-about 
and  complicated  way  of  treating  the  motion  of  a  planet, 
but,  as  a  matter  of  fact,  it  is  a  labor-saving  device; 
calculations  that  would  be  impossibly  long,  are  by  this 
method  reduced  to  workable  limits.  It  is  another  ex- 
ample of  the  old  adage,  that  the  longest  way  round  is 
the  quickest  way  home.  The  action  of  Jupiter,  as  here- 
tofore illustrated,  is  directly  upon  the  earth,  not  upon 
the  orbit,  for  the  orbit  is  purely  an  imaginary  con- 
ception— a  mathematical  fiction.  This  action  directly 
changes  the  speed  of  the  earth  and  the  direction  in 


The  Motions  of  the  Planets       131 

which  it  is  moving ;  and  this  change  in  speed  and  direc- 
tion can  be  computed.  With  such  corrected,  or  new, 
motion  of  the  earth,  a  new  orbit  could  be  computed, 
which  orbit  would  differ  very  slightly  from  the  original. 
It  might,  for  example,  be  slightly  larger  and  with  less 
curvature.  Thus  the  direct  effect  of  Jupiter  is  to  change 
the  actual  motions  of  the  earth;  the  indirect  effect  is  to 
change  the  so-called  orbit.  Now,  it  is  found  to  be 
easier  and  quicker  to  compute  these  indirect  changes 
upon  the  imaginary  orbit,  and  from  these  to  find  the 
actual  changes  in  the  earth's  position,  rather  than  to 
proceed  in  what  might  be  called  the  normal  straight  for- 
ward way. 

Another  advantage  of  this  indirect  method  becomes 
apparent  after  a  moment's  consideration.  The  action 
of  Jupiter  upon  the  earth  is  momentary,  but  the  effect 
upon  its  motions  is  permanent.  Suppose  that  the  earth 
were  the  sole  planet ;  its  path  would  then  be  elliptic  and 
constant — year  after  year,  the  earth  would  travel 
around  in  the  same  orbit.  Now,  if  Jupiter  were  sud- 
denly introduced  into  the  system,  as  at  J  in  Figure  20, 
it  would  pull  the  earth  out  of  its  path  and  enlarge  its 
orbit.  If  Jupiter  were  then  suddenly  removed,  blotted 
out  of  existence  as  it  were,  the  earth  would  go  on  for- 
ever traveling  in  its  new  path.  The  earth  could  never 
return  to  its  old  orbit.  Thus  the  effect  of  the  mo- 
mentary and  slight  disturbance  in  the  earth's  motion 
would  be  reflected  down  throughout  all  the  ages.  If 


Gravitation  versus  Relativity 

the  temporary  presence  of  Jupiter  increased  the  length 
of  the  year  by  one  second,  then  after  the  lapse  of  sixty 
years  the  earth  would  be  one  minute  late  in  returning 
to  E,  an  hour  late  after  the  lapse  of  3600  years,  and  a 
whole  day  late  after  86,400  years.  The  indirect  change 
in  the  orbit  thus  affords  a  ready  and  easy  means  of 
finding  the  effect  of  the  disturbance  at  some  distant 
future  date. 

The  astronomer,  therefore,  calculates  the  changes 
in  the  orbit  of  the  planet,  not  the  direct  changes  in  the 
motion  of  the  planet  itself.  The  methods  of  computing 
these  changes  in  the  orbit  of  a  planet  are  very  compli- 
cated ;  the  formulas  would  fill  hundreds  of  pages  of  this 
book  and  would  be  incomprehensible  to  the  ordinary 
reader,  few  physicists  or  mathematicians  even  have 
more  than  a  dim  idea  of  their  real  meaning,  or  the 
slightest  knowledge  of  how  to  use  them.  These  changes 
in  the  elements  of  a  planet's  orbit  are  called  "perturba- 
tions" or  "variations,"  and  one  may  thus  speak  of  the 
perturbation  of  the  major  axis  of  the  earth's  orbit,  the 
variation  of  the  perihelion  of  Mercury,  or  the  perturba- 
tion of  the  eccentricity  of  Jupiter's  orbit. 

In  calculating  these  perturbations,  the  mathematician 
is  forced  to  adopt  the  old  device  of  Hipparchus,  the 
discredited  and  discarded  epicycle.  It  is  true  that  the 
name,  epicycle,  is  no  longer  used,  and  that  one  may 
hunt  in  vain  through  astronomical  text-books  for  the 
slightest  hint  of  the  present-day  use  of  this  device, 


The  Motions  of  the  Planets      133 

which  in  the  popular  mind  is  connected  with  absurd 
and  fantastic  theories.  The  physicist  and  the  mathema- 
tician now  speak  of  harmonic  motion,  of  Fourier's 
series,  of  the  development  of  a  function  into  a  series  of 
sines  and  cosines.  The  name  has  been  changed,  but 
the  essentials  of  the  device  remain.  And  the  essential, 
the  fundamental  point  of  the  device,  under  whatever 
name  it  may  be  concealed,  is  the  representation  of  an 
irregular  motion  as  the  combination  of  a  number  of 
simple,  uniform  circular  motions.  It  is  so  necessary 
that  this  device  be  fully  understood,  that,  even  at  the 
risk  of  repetition,  a  fuller  illustration  is  given  in  Figure 
21. 

Suppose  C  to  be  moving  forward  in  the  direction 
of  the  arrow  at  a  uniform  speed,  while  the  particle 
A  revolves  in  a  circle  about  C,  as  a  centre.  As  C 
advances,  A  also  moves  forward  and  downward,  so 
that  when  C  has  moved  to  C',  A  has  made  a  quarter 
revolution  and  is  directly  in  advance  of  C'  at  D'.  Dur- 
ing this  interval,  the  entire  forward  motion  of  the  par- 
ticle A  has  been  equal  to  the  sum  of  the  forward  mo- 
tions of  C  and  of  A  in  the  circle :  the  downward  motion 
of  A  has  been  due  solely  to  its  motion  in  the  circle. 
During  the  next  quarter  revolution,  A  will  still  be  mov- 
ing downward,  but  will  be  moving  backward  instead  of 
forward,  and  its  actual  forward  motion  will  be  equal 
to  the  difference  between  the  forward  motion  of  C 
and  the  backward  motion  of  A.  At  the  completion 


134 


The  Motions  of  the  Planets      135 

of  one  half  revolution,  A  will  be  at  L,  directly  under 
the  then  position  of  the  centre,  C".  The  actual  path 
of  the  particle  is  shown  by  the  dotted  line,  AD'L, 
which  is  a  combination  of  the  straight  line  and  the 
circular  path. 

Now,  if  another  particle  B  be  supposed  to  revolve 
in  a  much  smaller  circle  about  A,  but  at  twice  the 
speed;  then,  in  a  similar  manner,  this  motion  in  the 
small  circle  can  be  combined  with  the  forward  motion 
of  C,  and  the  path  represented  by  the  dash-dot  line 
would  result.  The  actual  path  of  B,  however,  would 
be  the  combination  of  this  path  with  that  of  A,  or 
would  be  as  represented  by  the  full  heavy  line.  This 
actual  path  of  the  particle  appears,  at  first  glance,  to  be 
very  irregular,  but  it  is  clearly  made  up  of  the  combina- 
tion of  three  very  simple  motions — one  rectilinear,  the 
other  two  circular. 

It  is  not  necessary  to  plot  out  the  path  on  paper 
in  order  to  find  the  real  or  combined  motion  of  B. 
The  methods  and  formulas  of  ordinary  high-school 
trigonometry  furnish  a  ready  means.  It  will  be  re- 
membered that  for  any  angle,  such  as  M  in  the  dia- 
gram, the  ratio  of  the  base  CS  to  the  radius  CR  is 
called  the  cosine  of  the  angle,  and  the  ratio  of  the  alti- 
tude SR  to  the  radius  is  called  the  sine  of  the  angle, 
and  it  will  be  further  remembered  that  elaborate  tables 
are  at  hand  which  give  the  numerical  values  of  these 
ratios  for  angles  of  all  sizes.  Hence,  if  the  length 


136    Gravitation  versus  Relativity 

of  the  radius  CR  be  known,  then  for  any  particular 
value  of  the  angle  M, 

CS  —  CR  cosine  M 
SR  =  CR      sineM 

and,  by  taking  the  values  of  the  sine  and  cosine  from 
the  trigonometrical  tables,  these  simple  equations  en- 
able one  to  find  the  exact  numerical  values  of  CS  and 
SR.  But  the  particle  A  is  supposed  to  move  around  the 
circle  at  uniform  speed,  therefore  the  angle  M  increases 
uniformly  with  the  time.  Suppose  it  takes  just  one 
hour  for  the  particle  to  make  a  complete  circuit,  then 
in  one  minute  it  will  travel  6°,  or  it  will  take  10  seconds 
for  the  particle  to  move  over  a  single  degree,  and  thus 
the  exact  value  of  the  angle  can  be  found  for  any 
given  time.  The  length  of  time  for  the  particle  to 
travel  once  around  the  circle  is  called  the  "period,"  and 
the  radius  of  the  circle  is  called  the  "amplitude"  of 
the  motion. 

The  actual  distance  in  the  horizontal  direction  over 
which  the  particle  has  traveled  at  any  given  moment 
is  equal  to  the  distance  through  which  C  has  moved 
plus  the  value  of  CS  at  that  moment;  the  distance  of 
the  particle  above  the  central  line  is  equal  to  SR. 
Similarly  for  the  motion  of  the  particle  in  the  smaller 
circle,  the  formulas  of  trigonometry  give  its  position 
at  any  time.  In  the  particular  case  illustrated  in  the 


The  Motions  of  the  Planets      137 

diagram  it  was  supposed  that  B  made  just  two  revo- 
lutions while  A  made  one.  The  angle  which  represents 
its  motion,  therefore,  increases  just  twice  as  fast  as 
that  for  A;  or,  if  the  one  angle  be  M,  the  other  will 
be  2M. 

If  the  values  of  the  radii  of  the  two  circles,  or  the 
amplitudes  of  the  motions,  be  denoted  for  simplicity 
by  a  and  b,  respectively,  and  if  c  represents  the  rate 
at  which  the  centre  moves  forward  in  unit  time,  one 
minute  for  example,  then  will  the  position  of  B,  at 
any  instant,  be  given  by: 

Horizonal  position  =  c.t  +  a  cosine  M  +  b  cosine  2M 
Vertical  position  =  a  sine  M  +  b  sine  2.M 

From  these  two  expressions  any  one,  with  the 
slightest  knowledge  of  trigonometry,  can  find  the  exact 
position  of  B  at  any  instant.  But  this  position  can 
also  be  found  from  the  path  as  laid  out  in  our  dia- 
gram. The  one  method  of  finding  the  position  of  B 
requires  the  ability  to  use  tables  of  trigonometry,  the 
other  method  requires  the  ability  of  an  expert  draughts- 
man; the  one  method  reduces  the  path  of  the  particle 
to  the  form  of  a  mathematical  expression,  the  other 
visualizes  it  and  traces  it  as  it  actually  occurs  in  space. 
The  former  is  the  present  method,  known  under  vari- 
ous names;  the  latter  is  the  old  epicycle  method  of 
Hipparchus. 


138    Gravitation  versus  Relativity 

The  graphical,  or  mechanical  method  can  only  be 
used  when  the  motion  is  composed  of  very  few  and 
very  large  components.  In  simple  cases,  diagrams 
often  suffice,  and  actual  mechanical  devices  may  be 
made  to  represent  the  motion.  The  Tide  Predicting 
Machine  of  the  Coast  and  Geodetic  Survey  at  Wash- 
ington is  a  note-worthy  example  of  the  application  of 
the  mechanical  method.  The  rise  and  fall  of  the  tide 
at  any  port  is  a  periodic  phenomenon,  and  it  may, 
therefore,  be  analyzed,  or  separated  into  a  number  of 
simple  harmonic,  or  circular  components.  Each  com- 
ponent tide  will  be  simple,  will  have  a  definite  period 
and  a  constant  amplitude;  and  each  such  component 
may  be  represented  mechanically  by  the  arm  of  a 
crank,  the  length  of  which  represents  the  amplitude; 
each  crank  ami  being,  in  fact,  the  radius  of  one  of  the 
circles  in  our  diagram.  Such  a  machine  was  invented 
by  Sir  William  Thomson  and  was  put  in  operation 
many  years  ago.  The  machine  at  present  in  use  at 
Washington  was  designed  by  William  Ferrel.  It  pro- 
vides for  nineteen  components  and  directly  gives  the 
times  and  heights  of  high  and  low  waters.  In  order 
to  predict  the  tides  for,  a  given  place  and  year,  it  is 
necessary  to  adjust  the  lengths  of  the  crank  arms,  so 
that  each  shall  be  the  same  proportion  of  the  known 
height  of  the  corresponding  partial  tide,  and  to  adjust 
the  periods  of  their  revolutions  proportionally  to  the 
actual  periods.  Each  arm  must  also  be  set  at  the 


The  Motions  of  the  Planets      139 

proper  angle  to  represent  the  phase  of  the  component 
at  the  beginning  of  the  year.  When  all  these  adjust- 
ments have  been  made,  the  machine  is  started  and  it 
takes  only  a  few  hours  to  run  off  the  tides  for  a  year, 
or  for  several  years.  This  machine  probably  represents 
the  highest  possible  development  of  the  graphical  or 
mechanical  method.  It  is  a  concrete,  definite  mechani- 
cal adaptation  of  the  epicyclic  theory  of  Hipparchus. 
But,  because  the  Coast  Survey  represents  and  predicts 
the  movements  of  tidal  waters  by  a  complicated  mass 
of  revolving  cranks  and  moving  chains,  does  any  one 
imagine  for  a  moment  that  the  actual  waters  are  made 
up  of  such  a  system  of  cranks?  No  more  did  Hip- 
parchus believe  that  the  bodies  of  the  solar  system  were 
actually  attached  to  the  radial  arms  of  his  epicycles; 
his  was  a  mere  mathematical,  or  graphical  device  for 
representing  irregular,  complicated  motions. 

While  the  graphical,  or  mechanical  method  is  limited 
to  a  few  terms,  the  trigonometrical,  or  analytical 
method  is  unlimited.  It  is  possible  to  pile  epicycle 
upon  epicycle,  the  number  being  limited  only  by  the 
patience  of  the  mathematician  and  computer.  The  ex- 
pressions for  the  disturbing  action  of  one  planet  upon 
another,  due  to  the  attraction  of  gravitation,  involve 
an  unlimited  number  of  such  terms;  or,  as  the  mathe- 
matician puts  it,  the  series  is  infinite.  The  importance 
of  each  term  depends  primarily  upon  the  size  of  the 
coefficient,  or  amplitude  of  that  term — upon  the  length 


H°    Gravitation  versus  Relativity 

of  the  crank-shaft,  so  to  speak.  Now  these  coefficients, 
the  a's,  the  b's,  and  the  c's,  of  the  formulas,  are  them- 
selves extremely  difficult  to  calculate.  Each  one  is 
made  up  of  an  unlimited  number  of  separate  terms,  or 
is,  itself,  an  infinite  series.  The  principal  factor  in 
each  coefficient  is  the  ratio  of  the  respective  distances 
of  the  planets  from  the  sun,  and  the  size  of  the  coeffi- 
cient depends,  therefore,  primarily  upon  whether  this 
factor,  or  its  square,  or  its  third  power,  enters.  Nep- 
tune is  thirty  times  as  far  from  the  sun  as  is  the 
earth,  and  in  the  case  of  these  two  planets  this  factor 
is  i/3Oth.  The  square  of  this  is  1/900,  and  the  cube, 
or  third  power,  1/27000;  so  that  the  terms  decrease 
very  rapidly.  In  the  case  of  Mercury  and  Venus, 
however,  the  ratio  is  about  1/2,  and  the  powers  of 
this  factor  are  %>  H>  1/16,  etc.  These  diminish 
much  more  slowly  than  do  those  for  the  Earth  and 
Neptune,  with  the  consequent  result,  that  the  computa- 
tion is  very  much  longer  and  more  difficult. 

The  various  angles,  M,  which  represent  the  period 
of  each  term  in  the  expression  for  the  disturbed 
motion,  are  formed  by  various  combinations  of  the 
mean  motions  of  the  planets.  The  earth,  on  the 
average,  moves  about  one  degree  ( i  ° )  per  day ;  Mer- 
cury four  and  one-tenth  (4.1°)  degrees.  These  may 
be  combined  in  all  sorts  of  ways :  the  direct  difference 
between  them  is  3.1°;  twice  this  difference  is  6.2°; 
three  times,  9.3°.  Or  again  the  motion  of  Mercury 


The  Motions  of  the  Planets      141 

less  twice  that  of  the  earth  is  2.1°,  and  twice  this  is 
4.2°;  or  again  the  motion  of  Mercury  less  four  times 
that  of  the  earth  is  only  one-tenth  of  a  degree  0.1°. 
For  each  and  every  such  combination,  there  will  be  a 
corresponding  term  in  the  disturbed  motion  of  the 
planet.  The  period  of  the  perturbation  will  be  found 
by  dividing  360°  by  the  daily  motion.  For  the  relative 
motion  of  3.1°  per  day  the  period  is  about  116  days, 
for  the  relative  motion  of  0.1°  the  period  is  nearly 
ten  years. 

The  extreme  complexity  of  the  problem  may  be  best 
illustrated  by  giving  the  actual  expression  for  the  posi- 
tion of  the  Perihelion  of  Mercury,  as  affected  by  the 
action  of  Venus  alone.  This  is  taken  from  the  work 
of  Leverrier,  and  in  it  the  symbol  A  represents  the 
mean  motion  of  Mercury  and  1'  that  of  Venus.  The 
other  symbols  represent  various  elements,  or  combina- 
tions of  elements  of  the  two  planets. 

TABLE  I 

Perturbations  of  Mercury  by  Venus 

*  Perihelion  of  Mercury  =  75°  7'  I3".93    +  28o".6  t 

—  o".200  sin  (I'-X) 
-f  o".205  sin  2  (I'-X) 
+  o".i7i  sin  3  (I'-X) 
+  o".ii7  »w4(l'-X) 

etc. 

etc. 
46  other  terms 

*  Annales  de  1'Observatoire  de  Paris,  Mtmoires,  vol.  v. 


142    Gravitation  versus  Relativity 

Leverrier's  calculations  were  made  for  January  I, 
1850,  and  the  time,  t,  is  measured  in  centuries.  So 
that,  this  expression  means  that  on  January  i,  1850, 
the  perihelion  of  Mercury's  orbit  was  located  in  longi- 
tude 75°  7'  13". 93.  Due  to  the  action  of  Venus  alone, 
the  perihelion  was  then  moving  forward  at  a  rate  of 
280". 6  per  century.  This  uniform  forward  motion, 
however,  is  modified,  also  by  Venus,  by  fifty  periodic 
terms,  which  vary  in  size  from  4". 58  to  only  0^.015, 
and  with  corresponding  periods  of  a  few  days  to 
several  years.  In  order  to  find  the  actual  position  of 
the  perihelion  on  a  given  day,  February  25,  1921,  for 
example,  the  71  years  and  56  days,  which  elapsed  be- 
tween the  date  of  Leverrier's  epoch  and  the  date  in 
question,  must  be  reduced  to  the  equivalent  fraction 
of  a  century  (0.7115)  and  the  280". 6  multiplied  by 
this  figure.  The  result  is  I99".65,  or  3'  19" .65.  Then, 
from  the  known  values  of  A  and  1'  must  be  calculated 
the  exact  value  of  each  and  every  one  of  the  fifty 
various  angles :  the  sines  of  these  must  then  be  taken 
from  a  treatise  on  trigonometry,  and  each  sine  multi- 
plied by  the  corresponding  coefficient.  For  example, 
the  first  angle  is  5.1°,  the  sine  of  which  is  +  0.089 
Multiplying  this  by  the  coefficient,  —  o".2oo,  the  first 
term  becomes  on  February  25th,  —  o".oi8.  In  the 
same  way,  all  the  other  forty-nine  terms  are  calculated ; 
some  are  positive,  some  are  negative.  The  algebraic 
sum  of  all  these  terms  is  +  7". 49,  so  that  the  position 


The  Motions  of  the  Planets 

of   the   perihelion   on  February   25,    1921,    is   given 

by: 

Perihelion  =75°  /  i3"-93 
+  3  19  -65 
+  7  49 

75°  10'  4i".o7* 

It  may  be  of  interest  to  compare  this  position  with 
those  for  other  dates  in  the  same  year,  remembering 
that  the  figures  given  were  obtained  by  taking  account 
of  the  action  of  Venus  alone. 

1921,  February   25    75°io'4i".o7 
July  19  33".  10 

December  10  39"  45 

These  show  that  from  February  25  to  July  19 
the  perihelion  was  moving  backward,  while  during  the 
next  period  it  was  moving  forward,  but  on  December 
loth  it  was  still  behind  where  it  had  been  earlier  in 
the  year. 

All  this  is  complicated  enough,  but  it  only  accounts 
for  the  action  of  Venus;  it  requires  twenty-one  (21) 
similar  terms  to  account  for  the  action  of  the  earth, 
sixteen  (16)  for  Jupiter,  six  (6)  for  Saturn,  and  one 
(i)  for  Uranus. 

It  has  been  noted  above  that  each  one  of  the  coeffi- 

*  The  effects  of  Precession  and  of  the  other  planets  are  omitted : 
hence  this  value  will  not  agree  with  that  taken  from  the  Nautical 
Almanac. 


144    Gravitation  versus  Relativity 

cients  in  the  above  expression  for  the  position  of  the 
perihelion  is  itself  the  result  of  an  elaborate  calcula- 
tion. That  is,  the  figure  280". 6  for  the  secular  motion 
due  to  Venus  and  the  figure  o"  .200  for  the  first  periodic 
term,  both  depend  upon,  or  are  calculated  from,  infinite 
series,  which  series  involve  the  ratios  of  the  distances 
of  Mercury  and  Venus  from  the  sun,  together  with 
the  eccentricities  and  the  inclinations  of  their  respective 
orbits.  The  necessary  formulas  for  making  these  cal- 
culations are  given  by  Leverrier  in  the  Annals  of  the 
Paris  Observatory,  and  it  requires  fifty-seven  (57) 
quarto  pages  to  print  these  formulas  in  a  condensed 
and  symbolical  form.  That  is  Leverrier  uses  the 
symbol,  [i],  for  example,  to  represent  a  definite  long 
series,  which  is  printed  once  only,  while  the  symbol, 
or  abbreviation  may  enter  the  formulas  half  a  dozen 
or  more  times. 

The  perturbations  of  the  planets  fall,  as  has  been 
seen,  into  two  great  classes,  the  Periodic  and  the 
Secular. 

The  Periodic  Perturbations  are  those  which  involve 
the  trigonometrical  sines  and  cosines.  These  depend 
upon  the  relative  positions  of  the  planets  in  their 
respective  orbits;  they  are  mostly  small,  and  run 
through  their  courses  in  a  few  years,  or  a  few 
revolutions  of  the  planet  in  its  orbit. 

The  Secular  perturbations,  however,  are  of 
special  and  extreme  importance  in  all  theories  of 


The  Motions  of  the  Planets      H5 

planetary  motions.  These  are  the  terms  that  con- 
tain the  time  directly  as  a  factor,  similar  to  the  280". 6 
in  the  expression  for  the  perihelion  of  Mercury.  The 
presence  of  these  terms  indicate  permanent  changes  in 
the  orbits,  changes  progressing  steadily  with  the  time ; 
so  that,  after  the  lapse  of  centuries,  the  orbit  will  be 
completely  changed  in  character.  Advancing  at  the 
rate  of  280". 6  per  century  the  perihelion  of  Mercury 
would  make  one  complete  revolution  in  some  4600  cen- 
turies. Such  a  motion  of  the  perihelion  merely  changes 
the  position  of  the  orbit  in  space;  not  so,  however, 
with  changes  in  the  major  axes,  or  in  the  eccentricities 
of  the  orbits.  These  elements  determine  the  size  and 
shape  of  the  path  of  the  planet,  and  any  secular  changes 
in  these  elements  would  mean  the  ultimate  destruction 
of  the  solar  system. 

If  the  size  of  the  earth's  orbit  were  steadily  decreas- 
ing, it  would  mean  the  ultimate  collapse  of  the  earth 
into  the  sun;  if,  on  the  other  hand,  it  were  found  that 
the  orbit  were  uniformly  increasing  in  size,  then  would 
the  earth  be  finally  driven  from  the  solar  system  and 
would  disappear  into  endless  space.  Now  some  hun- 
dred years  ago  La  Place  showed  clearly  that  this  can 
never  happen:  there  are  no  secular  perturbations  of 
the  major  axis  of  any  orbit.  In  the  mathematical 
expression  for  the  length  of  the  axis  of  an  orbit  there 
is  no  term  corresponding  to  the  280". 6  in  the  motion 
of  the  perihelion.  The  distance  of  the  planet  from 

10 


Gravitation  versus  Relativity 

the  sun  is,  in  the  long  run,  constant;  it  suffers  slight 
periodic  variations,  but  always  comes  back  to  its  aver- 
age value  after  the  lapse  of  a  comparatively  short 
period  of  time. 

But  the  expression  for  the  eccentricity  of  an  orbit 
contains  a  secular  term.  This  apparently  indicates  a 
permanent  and  progressive  change  in  the  shape  of  the 
orbit.  The  eccentricities  of  Mercury,  Mars,  and  Jupi- 
ter are  increasing,  those  of  Venus,  the  earth,  Saturn, 
and  Uranus  are  decreasing.  According  to  Newcomb  * 
the  secular  changes  in  the  eccentricities  of  the  four 
inner  planets  are : 

Mercury  +  0.000,0206 

Venus  —  0.000,0470 

The  Earth  —  0.000,0416 

Mars  +  0.000,0907 

The  change  for  Mars  is  the  largest,  and  this  figure 
would  indicate  that,  after  the  lapse  of  some  ten  thou- 
sand centuries,  the  eccentricity  of  the  orbit  would  be 
unity,  and  the  orbit,  itself,  practically  a  parabola. 
Before  this  condition  could  be  reached,  however,  the 
orbit  would  be  so  narrow  that  the  planet  would  collide 
with  the  sun.  Such  secular  changes  in  the  eccentrici- 
ties, if  real,  indicate,  therefore,  the  final  and  complete 
collapse  of  the  solar  system. 

*  Astronomical  Constants,  page  109. 


The  Motions  of  the  Planets      147 

This  question,  as  to  the  Permanency,  or  Stability 
of  the  Solar  System,  has  been  investigated  by  the 
leading  mathematical  astronomers;  La  Place,  Lever- 
rier,  and  Tisserand,  to  name  a  few  of  the  most  eminent. 
They  have  carried  the  system  of  approximations  one 
step  further,  and  have  shown  that  in  this  next  step 
the  so-called  secular  perturbations  of  the  eccentricities 
and  inclinations,  at  least,  disappear  and  are  replaced 
by  periodic  terms  of  inconceivably  long  periods.  These 
periods  are  measured  by  the  thousands  of  years,  and 
depend  upon  the  relative  positions  of  the  orbits,  not 
upon  the  positions  of  the  planets  in  the  orbits.  The 
eccentricity  of  Mars,  for  example,  instead  of  steadily 
increasing,  will  increase  for  centuries,  then  decrease 
until  it  becomes  smaller  than  at  present,  then  turn  and 
again  become  larger.  These  fluctuations  are  confined 
within  comparatively  narrow  limits;  the  orbits  can 
never  become  circles,  nor  change  radically  from  their 
present  shapes.  La  Place  clearly  showed  that  the  orbits 
of  the  various  planets  are  so  bound  up  together  that, 
as  the  eccentricity  of  one  orbit  increases,  that  of  some 
other  orbit,  or  those  of  other  orbits,  must  correspond- 
ingly decrease. 

The  so-called  secular  perturbations  of  the  eccentrici- 
ties and  inclinations,  certainly,  do  not  exist;  they  are 
mathematical  fictions,  introduced  through  the  special 
methods  used  in  approximating  towards  the  true  values 
of  these  elements.  The  figures  given  for  such  secular 


Gravitation  versus  Relativity 

perturbations  in  all  text-books,  the  figures  found  in 
the  classic  works  of  Leverrier  and  of  Newcomb,  are 
mere  approximations  to  the  present  rate  of  change  of 
the  elements.  The  eccentricity  of  the  earth's  orbit  is, 
at  present,  decreasing  at  the  definite  rate  stated  by 
Newcomb,  but,  after  the  lapse  of  centuries,  this  rate 
will  change.  For  the  next  two  or  three  hundred  years, 
perhaps,  there  will  be  no  appreciable  change  in  this 
rate,  and  for  all  practical  purposes  of  today  it  is  suffi- 
ciently accurate  to  speak  of  this  rate  as  constant,  to 
speak  of  it  as  a  secular  perturbation.  But,  if  one 
wishes  to  consider  the  condition  of  the  solar  system, 
the  shapes  of  the  planetary  orbits,  say  ten  or  twenty 
thousand  years  ago,  then  these  so-called  secular  per- 
turbations must  be  discarded,  and  other  and  more 
accurate  methods  of  calculation  must  be  used. 

But  even  these  conclusions  of  La  Place  are  only 
approximate;  his  methods  are  only  a  second  or  third 
step  in  the  whole  series  of  successive  approximations 
towards  the  true  motions  of  the  various  planets. 
Leverrier,  as  the  result  of  an  extensive  research,  con- 
cludes that  it  is  impossible  to  determine  whether  the 
four  inner  planets,  Mercury,  Venus,  the  earth,  and 
Mars,  form  a  stable  system  or  not.  Tisserand  has 
confirmed  this  result,  and  has  specifically  warned  all 
students  of  Celestial  Mechanics  against  the  illusions 
of  the  stability  of  the  planetary  system. 

It  is  thus  seen  that  the  mathematical  astronomer  is 


The  Motions  of  the  Planets      149 

forced  to  trace  the  motions  of  the  planets  by  an 
elaborate  system  of  successive  approximations.  The 
first  step  is  comparatively  simple;  the  second  step  is 
intricate  and  complicated  to  an  almost  impossible 
degree;  further  steps  are  impracticable  to  complete. 
These  successive  approximations  or  steps  may  be  sum- 
marized as: 

1st.  Approximation: 

The  orbit  of  each  planet  is  considered  as  being 
an  unvarying  ellipse,  in  which  the  body  moves 
with  a  speed,  which  varies  in  a  definite  and  easily 
computed  manner.  This  speed  is  given  by 
Kepler's  second  law. 

This  approximation  is  sufficient  to  give  the 
positions  of  the  planets,  with  one  or  two  excep- 
tions, for  several  years  as  close  as  can  be  observed 
with  the  unaided  eye. 

2nd.  Approximation: 

The  orbit  of  each  planet  is  still  considered  as 
elliptical,  but  the  ellipse,  itself,  varies  in  size  and 
position. 

This  approximation  is  sufficient  for  all  practi- 
cal purposes  of  the  navigator  and  the  routine 
astronomer,  and  gives  the  positions  of  the  planets 
for  a  couple  of  hundred  years,  with  the  accuracy 
of  minor  telescopic  observations. 

This  takes  account  of  perturbations  of  the  first 
order;  considers  the  coefficients  of  all  the  terms 


15°    Gravitation  versus  Relativity 

as  actual  constants,  and  the  so-called  secular 
perturbations  as  real  and  as  progressing  uniformly 
with  the  time. 

3rd.  Approximation: 

Second  order  perturbations  are  included,  and 
the  secular  perturbations  are  shown  to  be  periodic 
in  character,  running  their  respective  courses  in 
immensely  long  periods  of  time. 

This  approximation  carries  the  mathematical 
development  to  the  highest  degree  possible,  and 
indicates  the  modifications  of  the  planetary  tables 
necessary  when  very  long  periods  of  time  are 
under  consideration.  It  involves  all  discussions 
as  to  the  permanency  of  the  solar  system,  and  as 
to  its  possible  evolution. 


CHAPTER  V 

THE  MOTION  OF  THE  PERIHELION  OF  MERCURY 

WHILE  MERCURY  is  a  comparatively  difficult  ob- 
ject to  observe  with  the  unaided  eye,  it  has  been  known 
from  pre-historic  times,  and  measurements  of  its  posi- 
tion in  the  heavens  run  back  as  far  as  the  second  century 
before  the  Christian  Era.  The  difficulty  of  observing 
the  planet  lies  in  its  nearness  to  the  sun. 

Mercury,  it  will  be  recalled,  is  the  nearest  planet  to 
the  sun,  and  revolves  about  that  central  body  in  an 
elliptic  orbit  at  an  average  distance  of  somewhat  less 
than  four-tenths  (0.38)  that  of  the  earth.  As  it 
travels  around  the  sun  in  this  curve,  the  planet  will, 
therefore,  appear  to  an  observer  on  the  earth  to  oscil- 
late backward  and  forward,  appearing  first  to  the  east- 
ward, then  to  the  westward  of  the  sun,  but  never  de- 
parting very  far  from  that  body.  When  it  is  to  the 
eastward  of  the  sun,  Mercury  may  be  seen  low  down 
in  the  western  sky  for  a  short  time  after  the  sun  has 
set :  when  it  is  to  the  westward  of  the  sun,  the  planet 
rises  before  that  body,  and  can  be  seen  in  the  early 
hours  of  the  morning.  On  account  of  the  long  twi- 
lights in  northern  latitudes,  the  planet  is  much  more 

151 


Gravitation  versus  Relativity 

difficult  to  observe  in  northern  countries  than  in  the 
lower  latitudes  of  the  Mediterranean,  where  the  science 
of  astronomy  had  its  birth.  Under  the  most  favorable 
conditions,  Mercury  may  be  seen  as  far  as  28°  from 
the  sun,  and  when  in  this  position  of  greatest  elonga- 
tion the  planet  appears  as  a  brilliant  star  of  the  first 
magnitude. 

Most  of  the  telescopic  observations  of  Mercury  are 
made  in  broad  daylight,  when  it  is  invisible  to  the  un- 
aided eye.  Such  observations  are  not  difficult  to  make, 
for  the  object-glass  may  be  screened  from  the  reflected 
glare  of  direct  sunlight.  And  these  daylight  observa- 
tions are  preferable  to  those  made  after  sunset,  for 
the  planet  can  be  observed  when  high  in  the  heavens 
and  away  from  the  mists  and  thick  atmosphere  that 
are  always  found  near  the  horizon.  But  with  all  possi- 
ble precautions,  daylight  observations  are  not  as  satis- 
factory as  those  made  at  night,  for  the  sun  heats  the 
air,  sets  up  currents,  and  produces  all  sorts  of  abnormal 
tremblings  and  refractions.  The  observations  of  Mer- 
cury, therefore,  are  not  as  accurate,  not  as  satisfactory 
as  those  of  Mars  or  of  Jupiter,  or  of  any  other  planet, 
which  can  be  observed  at  night. 

Due  to  the  greatly  varying  distances  between  the 
earth  and  Mercury  as  they  travel  their  respective  paths 
about  the  sun,  the  apparent  diameter  of  the  latter  planet 
varies  from  about  5"  to  13".  The  real  diameter  of  the 
planet  is  3000  miles,  or  about  ^jths  that  of  the  earth. 


The  Perihelion  of  Mercury       153 

The  path  of  the  planet  has  caused  considerable 
trouble  to  mathematical  astronomers.  This  is  largely 
due  to  the  great  eccentricity  of  its  orbit  and  to  its  high 
inclination,  but  it  has  also  been  due,  in  the  early  days 
of  astronomical  research,  to  the  difficulty  of  securing 
good  observations.  Kepler  made  tables  of  the  planet's 
motions  as  early  as  1627,  from  which  he  was  enabled 
to  predict  the  transit  of  the  planet  across  the  sun's  disc 
in  1631.  Halley  in  England  about  1680  and  Lalande 
in  France  about  one  hundred  years  later  computed 
tables,  which  were  far  more  exact  than  those  of  any 
preceding  astronomer.  These  two  sets  of  tables  were 
of  approximately  equal  accuracy,  as  was  evidenced  by 
the  transit  of  1786,  for  this  phenomenon  "took  place 
three-quarters  of  an  hour  later  than  the  time  fixed  for 
it  by  Lalande,  and  three-quarters  of  an  hour  earlier 
than  that  assigned  by  the  tables  of  the  English  astrono- 
mer." Thereafter,  Lalande  greatly  improved  his 
tables :  but  it  was  Leverrier,  who  finally  solved  the 
difficulties  and  who  in  1844  prepared  tables  of  wonder- 
ful precision. 

Leverrier  based  his  researches  upon  a  magnificent 
series  of  meridional  observations  of  the  planet  made  at 
the  Royal  Observatory  in  Paris.  This  series,  as  used 
by  Leverrier,  began  on  March  8,  1801,  and  extended  to 
August  1 8,  1842,  embraced  somewhat  over  four  hun- 
dred (400)  separate  observations  in  various  parts  of 
the  planet's  orbit,  and  covered  one  hundred  and  seventy- 


154    Gravitation  versus  Relativity 

two  (172)  successive  complete  revolutions  of  the  planet 
in  its  orbit.  He  calculated,  in  the  manner  outlined  in 
the  last  chapter,  the  periodic  and  secular  perturbations 
caused  by  each  of  the  six  planets,  Venus,  the  earth, 
Mars,  Jupiter,  Saturn,  and  Uranus.  The  development 
of  the  necessary  formulas  required  mathematical  ability 
of  the  highest  order;  the  labor  involved  in  the  numeri- 
cal calculations  was  enormous.  Fortunately  for  Lever- 
rier,  Napoleon  III  was  on  the  throne  of  France  and 
ample  funds  were  available  for  the  corps  of  skilled 
computers  necessary  to  carry  out  the  immense  task. 

Some  idea  as  to  the  intricacy  of  the  problem  and 
as  to  the  labor  involved  may  be  gathered  by  again  look- 
ing at  the  expression  for  the  motion  of  the  perihelion, 
as  given  on  page  141,  and  remembering  that  this  expres- 
sion involves  the  action  of  Venus  alone.  To  this  must 
be  added  terms  for  the  action  of  each  one  of  the  other 
planets,  and  when  this  is  done,  the  work  is  only  begun. 
For  similar  expressions  must  be  found  for  the  motion 
of  each  one  of  the  other  elements  of  Mercury's  orbit, 
for  the  eccentricity,  for  the  major  axis,  and  the  others. 
In  all,  Leverrier's  formulas  involved  the  use  of  231 
terms  to  express  the  periodic  perturbations  of  the 
longitude  and  118  terms  for  those  of  the  radius  vector. 
All  these  terms  must  be  sorted  out  and  collected  into 
tables,  so  that  the  position  of  the  planet  on  any  given 
date  can  be  determined.  From  such  preliminary  tables 
of  motion,  the  position  of  the  planet  was  determined 


The  Perihelion  of  Mercury       155 

for  the  exact  instant  on  which  each  one  of  the  four 
hundred  observations  was  made.  These  theoretical 
positions  were  then  compared  with  the  actual  observed 
places  of  the  planet,  and,  so  accurate  were  these  prelim- 
inary tables  of  Leverrier,  that  in  all  the  forty-one  years, 
the  greatest  deviation  between  the  predicted  and  the 
true  place  (heliocentric  longitude)  was  but  n". 

This  apparently  marvelous  agreement  between  theory 
and  observation  did  not  satisfy  Leverrier,  and  he  pro- 
ceded  to  correct  his  tables,  and  to  introduce,  as  further 
observations,  the  results  of  a  considerable  number  of 
transits.  These  transits  of  the  planet  across  the  solar 
disc  furnish  the  most  delicate  tests  of  any  tables  of 
motion,  and  locate,  with  extreme  accuracy,  the  exact 
position  of  the  planet  at  the  moment  of  observation. 
To  make  the  observation  no  delicate  measuring  instru- 
ment is  necessary:  a  telescope  large  enough  to  see  the 
planet  and  an  accurate  clock  are  all  that  are  required. 

At  intervals  of  about  116  days,  Mercury  passes  be- 
tween the  earth  and  the  sun.  If  the  orbits  of  the  two 
bodies  were  in  the  same  plane  then,  at  these  times  of 
inferior  conjunction,  Mercury  would  appear  to  pass 
centrally  across  the  disc  of  the  sun.  But  the  path  of 
Mercury  is  inclined  some  7°  to  the  ecliptic  and  hence, 
in  general,  the  planet  appears  to  pass  either  to  the 
north  or  to  the  south  of  the  sun,  as  shown  at  A  in 
figure  22.  In  this  position  the  planet  would  not  be 
visible,  even  with  the  aid  of  a  powerful  telescope,  as 


i56    Gravitation  versus  Relativity 

it  would  be  concealed  in  the  brilliant  glare  of  the  sun. 
But  when  the  conjunction  occurs  near  the  node,  N, 
in  the  diagram,  then  the  planet  passes  across  the  face 


A 

Fig.  22.    Transit  Limits. 

of  the  sun,  appearing  as  a  small  round,  black  spot. 
As  the  orbit  of  Mercury  intersects  that  of  the  earth  in 
two  points  in  opposite  parts  of  the  heavens,  there  are 
two  possible  places  where  transits  may  be  observed. 
The  sun  passes  the  ascending  node  of  Mercury's  orbit, 
as  shown  at  N,  on  November  Qth  of  each  year,  and  the 
opposite,  or  descending  node,  on  May  7th.  Transits 
can  only  happen,  therefore,  when  conjunctions  occur 
on  or  near  one  of  these  days.  The  limit  is  about  two 
days  for  May  and  about  four  days  for  November, 
so  that  November  transits  are  nearly  twice  as  numer- 
ous as  those  of  May.  During  the  last  century  there 
were  four  May  transits  and  nine  in  November. 

The  observation  consists  in  noting  the  exact  instant 
at  which  the  disc  of  the  planet  becomes  tangent  to  the 
edge  of  the  sun ;  and  this  instant  can  easily  be  de- 
termined to  within  a  few  seconds  of  time.  There  are 


The  Perihelion  of  Mercury       157 

four  such  instants  of  contact,  as  will  be  noted  from  the 
accompanying  figure:  two  external  and  two  internal. 
The  internal  contacts  can  be  observed  with  greater 


Fig.  23.    External  and  Internal  Contacts. 

accuracy  than  the  external  ones.  The  first  external 
contact  is  the  most  difficult,  for  it  must  be  remembered 
that  the  planet  is  invisible  as  it  approaches  the  sun, 
and  does  not  become  visible  until  it  has  advanced 
sufficiently  far  upon  the  solar  disc  as  to  make  a 
decided  black  nick  in  the  edge.  There  is  some 
difficulty  also  with  the  internal  contacts  on  account 
of  irradiation  and  defects  in  telescopic  definition.  But 
with  all  these  possible  defects,  a  transit  serves  to 
determine  the  position  of  Mercury  in  its  orbit  with 
great  accuracy.  And  such  observations  were  of 
especial  value  before  the  year  1800,  when  the  long 
series  of  meridian  observations  were  begun  in  Paris. 
Before  that  date,  observations  of  the  planet,  other  than 
transits,  were  so  crude  as  to  be  of  little  or  no  value  to 
Leverrier  in  checking  his  tables. 


Gravitation  versus  Relativity 

The  first  transit  ever  observed  was  that  predicted 
by  Kepler,  and  occurred  on  November  7,  1631.  This 
was  observed  at  Paris  by  Gassendi,  who  in  a  letter 
gave  the  following  interesting  account:  "The  crafty 
god,"  says  he,  "had  sought  to  deceive  astronomers  by 
passing  over  the  sun  a  little  earlier  than  was  expected, 
and  had  drawn  a  veil  of  dark  clouds  over  the  earth  in 
order  to  make  his  escape  more  effectual.  But  Apollo, 
acquainted  with  his  knavish  tricks  from  his  infancy, 
would  not  allow  him  to  pass  altogether  unnoticed.  To 
be  brief,  I  have  been  more  fortunate  than  those  hunters 
after  Mercury,  who  have  sought  the  cunning  god  in 
the  sun.  I  found  him  out,  and  saw  him,  where  no  one 
else  had  hitherto  seen  him."  This  transit  occurred 
4h.  49m.  305.  in  advance  of  the  time  predicted  by 
Kepler.  Unfortunately,  the  times  given  by  Gassendi 
are  not  sufficiently  accurate  to  allow  Leverrier  to  make 
use  of  this  observation  in  correcting  his  tables. 

The  first  transit,  which  was  observed  with  all  the 
necessary  accuracy,  was  that  of  November,  1697.  This 
enabled  Leverrier  to  check  his  tables  with  observations 
extending  over  a  period  of  145  years,  or  60 1  complete 
revolutions  of  the  planet  in  its  orbit.  In  all  he  made 
use  of  nine  (9)  November  transits  and  five  (5)  May 
transits;  the  last  one  to  be  included  in  his  work  was 
that  of  November  10,  1848. 

After  correcting  his  preliminary  elements  and  tables 
of  motion,  Leverrier  made  a  careful  comparison  of  the 


The  Perihelion  of  Mercury       159 

theoretical  positions  derived  from  his  tables  with  the 
actual  positions  as  given  by  the  four  hundred  meridian 
observations  and  the  fourteen  transits.  A  very  curi- 
ous fact  developed.  There  were  no  serious  discrepan- 
cies between  the  theoretical  and  observed  places  so  far 
as  the  four  hundred  meridian  observations  were  con- 
cerned; nor  were  there  any  large  errors  furnished  by 
the  nine  November  transits.  All  these  observations 
were  completely  satisfied  and  explained  by  Leverrier's 
theoretical  orbit,  as  modified  by  the  calculated  pertur- 
bations caused  by  the  other  planets.  Not  so,  however, 
with  the  five  May  transits.  These  five  observations, 
between  1753  and  1845,  showed  discordances  between 
theory  and  observation,  which  Leverrier  deemed  to  be 
inadmissibly  large.  To  quote  his  own  words : 

"Leaving  aside  the  observations  of  1661  and  1677,  it 
may  be  noted  that,  first  the  observations  of  the  transits 
near  the  ascending  node  (November)  give  but  small 
errors;  while  the  transits  near  the  descending  node  (May) 
give  rise  to  an  error  of  12 ".05  in  1753,  which  diminishing 
nearly  regularly  with  the  time,  reduces  to  -  i".O3  in 
1845.  These  thirteen  seconds  in  92  years  require  to  be 
taken  into  serious  consideration" 

"They  cannot  be  attributed  to  errors  of  observation  of 
the  transits,  for  that  is  to  suppose  that  all  astronomers 
made  considerable  errors  in  the  measures  of  the  times  of 
contact:  these  errors,  moreover,  must  vary  in  a  pro- 
gressive manner  with  the  time,  and  must  differ  by  several 
minutes  at  the  end  of  the  period  of  92  years.  Circum- 
stances wholly  inadmissible!'' 


160    Gravitation  versus  Relativity 

Now  let  us  see  just  what  these  wholly  inadmissible 
errors  really  are.  In  92  years  Mercury  made  381  com- 
plete revolutions  in  its  orbit  about  the  sun  and,  during 
this  time,  travelled  some  trillions  of  miles,  and,  at  the 
end  of  the  period,  was  out  of  place,  according  to  Lever- 
rier's  tables,  by  some  2300  miles,  by  about  two-thirds 
(2/3)  of  its  own  diameter!  If  in  addition  to  the 
real  Mercury  we  conceive  of  an  imaginary  body  of  the 
same  size  as  the  real  planet  traveling  about  the  sun  in 
the  theoretical  path  computed  by  Leverrier,  then,  dur- 
ing the  92  years,  the  real  and  the  imaginary  bodies 
would  never  have  been  completely  separated.  At  the 
end  of  the  period,  the  two  bodies,  the  real  planet  and 
the  imaginary  one,  would  still  be  overlapping  by  one- 
third  of  their  diameters.  The  disc  of  Mercury,  itself, 
is  too  small  ever  to  be  seen  by  the  unaided  eye,  and 
it  would  thus  take  several  centuries  for  the  distance 
between  the  real  and  imaginary  planets  to  become  great 
enough  for  the  two  bodies  to  be  seen  as  separate  and 
distinct.  When  the  intricate  formulas  are  considered 
and  the  enormous  numerical  calculations  remembered, 
the  wonder  is,  not  that  there  should  be  this  slight  dis- 
crepancy in  Leverrier's  calculated  places,  but  that  the 
discordance  should  be  so  infinitely  minute.  The  genius 
of  Leverrier  as  a  mathematician,  and  his  patience  and 
industry  as  a  computer  cannot  be  too  highly  eulogized. 

In  correcting  his  elements  and  adjusting  his  tables  to 
conform  to  this  minute  discrepancy  in  motion,  Lever- 


The  Perihelion  of  Mercury       161 

rier  was  confronted  by  the  difficulty  already  alluded 
to:  the  discrepancy  appeared  in  observations  made  in 
only  one  part  of  the  orbit.  Observations  made  in  all 
other  parts  of  the  planet's  path  were  satisfactorily  ac- 
counted for.  Had  this  not  been  so,  the  discrepancy 
could  very  well  have  been  accounted  for  by  a  minute 
correction  in  the  mean  motion  of  the  planet,  by  chang- 
ing the  length  of  its  year  by  a  very  small  fraction  of 
a  second  of  time.  Such  a  change,  however,  in  the 
average  speed  of  the  planet  about  the  sun  would  spread 
the  discrepancy  all  around  the  orbit  and  it  would  appear 
proportionately  in  all  the  observations.  The  discord- 
ance, as  found  by  Leverrier,  could  only  be  explained 
by  some  combination  of  motions,  which  counter- 
balanced, or  neutralized  each  other  in  every  part  of  the 
orbit,  except  near  the  descending  node.  The  correc- 
tions to  the  elements,  used  in  forming  the  tables,  must 
be  such  that  they  nearly  destroy  each  other  for  the 
November  transits,  while  their  effects  are  added  to- 
gether for  the  May  transits.  Leverrier  found  such  a 
combination  in  changes  to  the  secular  perturbations  of 
the  perihelion  and  of  the  eccentricity.  From  the  two 
sets  of  transits  he  deduced  the  relation  between  the 
necessary  corrections  to  the  annual  motion  of  the  peri- 
helion, n',  and  that  of  the  eccentricity,  e' ;  namely 

2.72  e'  +  n'  =  o".392  * 

*  "Annales   de   TObservatoire   de   Paris,"     Memoires,  vol.   v, 
page  78- 


XI 


1 62    Gravitation  versus  Relativity 

Any  values  of  n'  and  of  e'  that  will  satisfy  this  equa- 
tion will  also  account  for  the  discrepancies  in  the 
transit  observations,  A  correction  to  the  calculated 
value  of  the  secular  motion  of  the  perihelion  thus  con- 
notes a  similar  change  or  correction,  in  the  computed 
secular  term  of  the  eccentricity.  From  the  observa- 
tions, alone,  it  is  impossible  to  determine  the  individual 
values  of  e'  and  n' ':  the  numerical  value  of  the  total 
combination  only  can  be  fixed. 

Now,  according  to  Leverrier's  calculations,  the  entire 
secular  motion  of  the  perihelion  amounts  to  526".?  in  a 
century,  or  to  5 ".267  per  annum,  while  the  correspond- 
ing secular  changes  in  the  eccentricity  are  4"  .2  per 
century,  or  0^.042  per  annum.  Any  large  change  in 
the  term  of  the  eccentricity  is  precluded  by  the  very 
size  of  the  motion  to  be  corrected  and  by  other  con- 
siderations. Leverrier  finally  concluded  that  by  far 
the  greater  portion  of  the  necessary  correction  must 
be  made  in  the  motion  of  the  perihelion.  A  reason  for 
this  is  easily  seen,  for  if  the  computed  secular  motion 
of  the  perihelion  (5 ".267)  and  of  the  eccentricity 
(o".O42)  be  combined  in  a  manner  similar  to  the  equa- 
tion above  given,  then  the  numerical  value  of  such  com- 
bination would  be  5 ".38 1.  But  the  correction  to  this 
was  found  to  be  o".392,  or  a  trifle  over  7  per  cent. 
And  the  corresponding  percentage  of  the  whole  motion 
of  the  perihelion  is  about  0^.38  per  annum :  this  is  the 
value  of  the  correction  finally  adopted  by  Leverrier. 


The  Perihelion  of  Mercury       163 

There  will,  of  course,  be  a  corresponding  correction  to 
the  eccentricity,  which  can  be  found  from  the  equation, 
and  which  amounts  to  about  10  per  cent  of  the  original 
computed  value.  To  again  quote  from  Leverrier : 

"The  necessity  for  a  considerable  increment  to  the 
secular  motion  of  the  perihelion  results  exclusively  from 
the  observations  of  the  transit  of  the  planet  across  the 
disc  of  the  sun:  further  only  times  of  internal  contacts* 
which  can  be  observed  with  great  accuracy,  were  used.  To 
escape  from  this  necessity,  it  must  be  admitted  that  errors 
of  several  minutes  in  the  estimation  of  the  time  of  this 
phase  must  have  been  made  in  the  large  observatories,  for 
example  in  1743  or  in  1753  at  Paris,  and  by  observers 
such  as  La  Caille,  de  Lisle,  Bouguer,  the  Cassinis. — An 
unacceptable  hypothesis!" 

A  possible  explanation  of  the  discrepancy  might  lie 
in  the  use  of  erroneous  masses  for  the  planets  in  all 
the  computations  upon  which  the  perturbations  de- 
pend. The  mass  of  a  planet,  which  has  a  satellite,  as 
Jupiter  for  example,  can  be  very  accurately  determined ; 
but  the  determination  of  the  mass  of  a  planet,  without 
a  satellite,  is  one  of  the  most  difficult  problems  of  celes- 
tial mechanics.  The  masses  of  Mercury  and  Venus 
can  only  be  found  through  the  effects  of  their  attrac- 
tions upon  other  bodies;  that  of  Venus,  through  the 
perturbations  caused  in  the  motions  of  Mercury.  The 
mass  of  Venus  is  a  direct  factor  in  the  computed  value 
of  the  annual  motion  of  the  perihelion  of  Mercury,  and 


1 64    Gravitation  versus  Relativity 

this  mass  is  uncertain.  In  his  computations  Leverrier 
used  the  value  1/401,800  for  the  mass  of  Venus  [that 
of  the  sun  being  unity],  and  with  this  value  he  found 
that  Venus  caused  280". 6  out  of  the  total  motion  of 
the  perihelion  of  526". 7  per  century.  To  be  the  cause 
of  the  38"  excess  motion,  the  mass  of  Venus  would 
have  to  be  increased  in  the  proportion  of  38"  to  281", 
or  by  practically  one-seventh  (1/7).  Of  course  this 
increase  in  the  mass  of  Venus,  necessary  to  account  for 
the  excess  motion  of  the  perihelion,  would  increase  in 
like  proportion  all  the  perturbations  of  the  other  ele- 
ments of  Mercury  and  those  of  the  other  planets  as  well. 
And  such  an  increase  would  cause  many  discrepancies : 
discordances  far  worse  than  that  in  the  motion  of  the 
perihelion.  It  thus  appears  very  clearly  that  the  peri* 
helial  motion  cannot  be  due  to  the  use  of  an  erroneous 
value  of  the  mass  of  Venus,  and  its  cause  must  be 
sought  elsewhere. 

Leverrier  found  the  most  plausible  explanation  of  the 
observed  excess  in  the  motion  of  the  perihelion  to  be 
the  presence  of  a  planet,  or  a  group  of  planets,  be- 
tween Mercury  and  the  sun.  He  announced  the  results 
of  his  calculations  to  the  Academy  of  Sciences  on  Sep- 
tember 12,  1859,  and  showed  that  a  body,  about  the 
size  of  Mercury  itself  and  revolving  about  the  sun  at 
a  distance  about  midway  between  that  luminary  and 
Mercury,  would  so  act  upon  Mercury  as  to  fully  ac- 
count for  the  unexplained  motion  of  the  perihelion.  If 


The  Perihelion  of  Mercury       165 

the  body  were  nearer  the  sun,  its  mass  would  have  to 
be  greater;  if  nearer  Mercury,  its  mass  would  be  less. 
While  such  a  body  could  never  be  seen  under  ordinary 
circumstances,  it  ought  to  be  conspicuous  at  the  times 
of  total  solar  eclipses,  and  it  might  be  seen  in  transit 
across  the  solar  disc.  Leverrier  discussed  the  possi- 
bility of  such  a  body  being  observed,  and  called  atten- 
tion to  the  fact  that  a  ring  of  asteroids  would  pro- 
duce the  same  effect  upon  the  motion  of  Mercury  and 
might  easily  escape  detection  in  the  glare  of  the  sun. 

Following  almost  immediately  upon  the  publication 
of  this  paper  came  the  announcement  that  such  a  planet 
had  actually  been  observed  in  transit  across  the  sun's 
disc  on  March  26,  1859,  by  Dr.  Lescarbault.  Lever- 
rier investigated  this  reputed  discovery,  and  was  con- 
vinced of  its  substantial  accuracy.  He  named  the  new 
planet  "Vulcan,"  and  computed  elements  of  its  orbit. 
Several  similar  observations  were  afterwards  reported, 
and  the  planet  was  even  detected  on  the  Greenwich 
photographs  of  the  sun.  From  the  elements  of  the 
orbit  Leverrier  predicted  the  dates  of  future  transits. 
On  the  days  set,  however,  no  object  was  seen  against 
the  solar  disc,  which  could,  by  any  possibility,  be  a 
planet  in  transit.  Much  doubt  was  thrown  upon  the 
original  observation  of  Dr.  Lescarbault,  and  several 
of  the  other  reputed  planets  in  transit  were  shown  to 
have  been  sun-spots. 

Fresh  interest  was  aroused  in  "Vulcan,"  when  two 


1 66    Gravitation  versus  Relativity 

well  known  American  astronomers  rediscovered  it  dur- 
ing the  total  solar  eclipse  of  1878.  Watson  and  Swift 
each  claimed  to  have  seen  in  the  vicinity  of  the  obscured 
sun,  not  one,  but  two  planets.  Yet,  neither  one  of  the 
four  supposed  planets  could  be  identified  with  Lever- 
rier's  "Vulcan,"  which,  if  real,  must  have  been  at  that 
moment  on  the  other  side  of  the  sun.  Astronomers 
finally  came  to  the  conclusion  that  both  Swift  and 
Watson  had  been  mistaken  in  their  observations,  and 
had  really  seen  two  fixed  stars,  not  planets. 

But  for  several  eclipses  the  search  for  intra-Mer- 
curial  planets  was  one  of  the  principal  objects  of  every 
expedition.  During  the  few  moments  of  totality,  lead- 
ing astronomers  carefully  examined  the  vicinity  of  the 
sun  for  suspicious  objects;  photography  was  brought 
into  service.  No  planet  has  been  found,  and  belief  in 
Vulcan  has  ceased  to  exist.  All  the  evidence  to-day 
would  seem  to  show  conclusively  that  there  is  not, 
within  the  orbit  of  Mercury,  a  body  sufficiently  large  to 
be  detected,  nor  large  enough  to  account  for  the  pecu- 
liar motions  of  Mercury. 

While  it  is  thus  now  known  that  "Vulcan"  does  not 
exist,  yet  the  second  suggestion,  that  of  a  group  or 
ring  of  planetoids  near  the  sun,  finds  more  and  more 
confirmation.  This  suggestion  was  deemed  by  Lever- 
rier,  himself,  to  be  extremely  probable,  but  it  was  lost 
sight  of  in  the  excitement  caused  by  the  reputed  dis- 
coveries of  a  single  planet. 


The  Perihelion  of  Mercury       167 

Some  twenty  years  after  the  classic  papers  of  Lever- 
rier  were  published,  Newcomb,  in  Washington,  under- 
took an  extension  and  revision  of  Leverrier's  work. 
Four  transits  had  occurred  in  the  meantime — three  in 
November  and  one  in  May — and  it  appeared  of  extreme 
importance  to  include  these  observations  in  the  test. 
The  entire  work  was  based  upon  that  of  Lever rier, 
whose  intricate  calculations  of  the  motions  and  per- 
turbations of  Mercury  were  taken  as  correct:  correct 
theoretically  and  numerically.  Before  this  assumption 
was  made,  however,  one  portion  of  the  work  was 
checked.  George  W.  Hill,  the  eminent  mathematician 
of  the  American  Ephemeris  and  Nautical  Almanac,  re- 
computed the  secular  perturbations  of  Mercury  caused 
by  Venus,  and  used  formulas  and  methods  of  compu- 
tation quite  different  from  those  of  Leverrier.  Hill's 
results  agreed  remarkably  with  those  of  Leverrier :  the 
centennial  variation  of  the  perihelion  being  according 
to  Hill  280". 50  instead  of  the  280^.64  found  by 
Leverrier.  The  difference  between  these  results  is  con- 
siderably less  than  one-tenth  of  one  per  cent  (0.05%) 
of  the  computed  value,  and  is  less  than  i/27oth  of  the 
38"  discrepancy.  It  would  thus  seem  to  be  clearly  es- 
tablished that  Leverrier's  classic  work  is  substantially 
correct,  both  theoretically  and  numerically. 

Newcomb*  thoroughly  investigated  each  and  every 

*  Astronomical  Papers  of  the  American  Ephemeris  and  Nautical 
Alanac.    Vol.  i:  1882. 


1 68    Gravitation  versus  Relativity 

one  of  the  transits  observed  between  the  years  1631 
and  1 88 1.  He  collected  all  the  recorded  observations, 
making  use  of  many  that  had  been  overlooked  or  dis- 
carded by  Leverrier.  It  will  be  remembered  that  Lever- 
rier  took  account  of  the  internal  contacts  only ;  New- 
comb,  on  the  other  hand,  utilized  all  four  contacts, 
wherever  possible,  in  each  observed  transit.  In  some 
instances,  of  course,  one  or  more  of  the  contacts  were 
wholly  unobserved,  due  to  clouds  in  the  sky,  or  to  the 
sun  being  below  the  horizon  at  the  moment.  A  number 
of  the  individual  observations  were  rejected  as  being 
inconsistent  with  the  majority  made  at  the  time;  and 
some  observations,  when  made  under  obviously  poor 
conditions,  were  given  small  weight.  Newcomb  pains- 
takingly investigated  each  and  every  dubious  observa- 
tion and  tried  to  obtain,  from  the  many  observations 
made  at  each  transit,  the  time  which  most  correctly 
represents  the  actual  occurrence.  Some  idea  of  the 
difficulties  encountered  may  be  had  by  reference  to 
the  transit  of  1753.  Seventeen  observations  of  external 
contact  are  cited  by  Newcomb;  the  difference  between 
the  earliest  and  the  latest  time,  as  given  by  the  individ- 
ual observers  was  67  seconds.  Eleven  of  these  obser- 
vations were  discarded  by  Newcomb  as  being  either 
clearly  erroneous  or  of  dubious  value.  The  six  re- 
maining observations  differed  among  themselves  by 
22  seconds.  Newcomb  took  the  mean  of  these  six  as 
representing  the  real  time  at  which  the  phenomenon 


The  Perihelion  of  Mercury       169 

took  place.  Similarly  in  the  transit  of  1799,  the  indi- 
vidual recorded  times  of  external  contact  differed  by 
i  minute  and  41  seconds.  After  rejecting  several  of 
these  as  probably  worthless,  there  remained  twelve  ob- 
servations, which  differed  among  themselves  by  58 
seconds.  The  mean  of  these  twelve  was  taken  by  New- 
comb  as  the  observed  time  of  contact,  and  this  mean 
was  found  by  him  to  differ  from  the  theoretical  or 
calculated  time  by  only  34  seconds.  Some  of  the  actual 
observations,  thus,  differed  by  only  three  or  four  sec- 
onds from  the  time  computed  theoretically  from  Lever- 
rier's  tables. 

To  show  how  difficult  it  is  to  determine  the  actual 
time  at  which  a  contact  actually  took  place,  let  us  com- 
pare the  times  of  internal  contact  in  the  transit  of 
1845,  as  determined  by  Newcomb  and  Lever rier  re- 
spectively. The  latter  astronomer  used  ten  observa- 
tions, made  by  the  best  observers  in  Europe,  which 
differed  among  themselves  by  eleven  seconds.  New- 
comb  used  twenty-five  observations,  several  of  which 
were  made  in  America.  Two  of  those  used  by  Lever- 
rier  he  discarded,  so  that  his  twenty-five  observations 
consisted  of  eight  used  by  Leverrier  and  seventeen  that 
were  apparently  not  available  at  the  time  the  French 
astronomer  completed  his  work.  Each  astronomer  took 
the  mean  of  the  observations  used;  Leverrier  of  his  ten, 
and  Newcomb  of  his  twenty-five.  The  results  differed 
by  six  (6)  seconds.  In  other  words,  disregarding  all 


J7°    Gravitation  versus  Relativity 

theoretical  considerations,  all  hypothetical  orbits  and 
formulas  of  celestial  mechanics,  and  considering  solely 
the  observations  of  reputable  astronomers,  Newcomb 
fixes  the  time  of  contact  in  the  transit  of  1845  B*x  sec~ 
onds  earlier  than  Leverrier.  This  may  seem  to  be  an 
almost  negligible  quantity,  but  the  entire  discrepancy 
between  theory  and  observation,  at  that  moment, 
amounted  to  only  nineteen  (19)  seconds.  The  differ- 
ence between  Newcomb  and  Leverrier  as  to  the  actual 
observed  time  of  the  phenomenon  amounts,  thus,  to 
nearly  one-third  (l/z)  of  the  entire  discordance  to  be 
explained. 

However  much  the  two  researches  may  differ  in  de- 
tail, the  general  results  are  in  full  and  complete  accord. 
Newcomb  found  that  the  discordance  between  the  ob- 
served and  theoretical  motions  of  the  perihelion  of 
Mercury,  as  pointed  out  by  Leverrier,  really  exists  and 
is,  in  fact,  slightly  larger  than  he  supposed.  Newcomb 
gives  this  excess  motion  as  43"  per  century,  as  against 
the  38"  found  by  the  eminent  French  astronomer. 

There  is  one  element  of  uncertainty  in  all  these  con- 
clusions, and  that  element  is  the  mass  of  Venus.  The 
size  of  the  perturbations  of  Mercury  depend  upon  this 
mass  of  Venus,  and  the  mass,  itself,  can  only  be  found 
from  the  perturbations.  There  is  here  somewhat  of  a 
vicious  circle.  But,  the  mass  can  be  found,  not  only 
from  the  motion  of  the  perihelion,  but  also  from  the 
motion  of  the  node  and  from  a  number  of  the  periodic 


The  Perihelion  of  Mercury       171 

inequalities.  Newcomb  investigated  this  question  and 
determined  the  mass  of  Venus  in  a  number  of  different 
ways,  using,  of  course,  in  each  case  the  motions  as 
actually  observed.  Some  of  these  results  are  shown 
in  the  following  table : 


From  the  perihelion  of  Mercury 


From  the  node  of  Mercury 
From  the  periodic  inequalities 


396,000 


the  mass  of  the  sun  being  unity.  Every  method 
of  determining  the  mass  of  Venus,  except  that 
of  the  motion  of  the  perihelion,  leads,  thus, 
to  a  figure  approximating  closely  to  1/400,000,  and 
this  exceptional  figure  differs  by  more  than  ten  (10%) 
per  cent  from  all  the  others.  There  is  a  decided  pre- 
ponderance of  evidence  in  favor  of  the  figure 
1/400,000.  Further,  if  the  mass  of  Venus  were  taken 
as  being  1/347,800  so  as  to  fully  account  for  the 
motion  of  the  perihelion,  large  discrepancies  would  be 
at  once  introduced  into  the  motion  of  the  node  and 
into  many  of  the  larger  periodic  perturbations.  The 
accumulation  of  evidence,  to-day,  points  rather  to  a 
smaller  mass  for  Venus;  the  accepted  value  being 
1/408,000,  or  practically  the  same  as  that  derived  from 
the  motion  of  the  node. 


Gravitation  versus  Relativity 

Thus  it  follows,  that,  with  the  best  attainable  values 
of  the  masses  of  the  planets,  the  calculated  motion  of 
the  perihelion  of  Mercury  is  less  than  the  observed 
value,  and  this  discrepancy  must  be  considered  as  real. 

Newcomb  continued  his  researches  on  the  motions 
of  the  planets,  redetermining,  from  all  available  mate- 
rial, the  elements,  masses,  and  motions  of  the  four 
inner  planets.  He  used  Leverrier's  tables  as  the  basis 
of  his  work,  and  determined  the  corrections  to  Lever- 
rier's elements,  which  would  best  satisfy  all  the  con- 
ditions furnished  by  the  observations.  The  entire 
work  was  inter-locking  and  the  labor  involved  enor- 
mous. The  number  of  meridian  observations  of  the 
various  planets  actually  included  in  the  investigation 
was: 

Of  the  Sun  40,176 

Mercury  5»42i 

Venus  12,319 

Mars  4>H4 

Total  62,030 

In  addition  there  were  included  all  the  transits  of  Mer- 
cury and  of  Venus,  involving  many  more  hundreds  of 
separate  observations. 

As  the  numerical  values  of  the  perturbations  de- 
pend directly  upon  the  masses  of  the  planets,  New- 
comb  made  a  new  and  exhaustive  investigation  and  re- 
determination  of  the  masses  of  all  the  principal  planets. 
That  of  Jupiter  was  found  from  the  motions  of  its 


The  Perihelion  of  Mercury       173 

satellites,  from  its  action  upon  Saturn,  upon  two  of 
the  planetoids,  and  upon  a  couple  of  comets;  that  of 
Mars  from  the  motions  of  its  satellites;  and  that  of 
the  earth  from  the  many  and  varied  measures  of  the 
solar  parallax.  The  masses  of  these  three  planets  were 
thus  obtained  from  independent  sources  and  are  known 
with  a  relatively  high  degree  of  accuracy.  In  deter- 
mining the  masses  of  Mercury  and  Venus,  Newcomb 
used  the  periodic  perturbations  solely,  so  that  the  mass 
of  Venus  is  derived  from  the  periodic  perturbations  of 
Mercury  and  of  the  earth,  produced  by  its  action;  the 
mass  of  Mercury  is  derived  from  the  periodic  irregu- 
larities in  the  motions  of  Venus.  Thus  the  masses  of 
all  the  planets,  as  used  by  Newcomb,  are  entirely  inde- 
pendent of  the  secular  motions  of  the  elements;  inde- 
pendent of  the  motion  of  the  perihelion  of  Mercury. 

As  this  classic  and  monumental  research  neared  its 
completion,  Newcomb  found  secular  variations  of  the 
elements,  besides  that  of  the  perihelion  of  Mercury, 
which  could  not  be  satisfactorily  represented  by  the 
equations  and  formulas  of  celestial  mechanics.  In  a 
note,  published  in  the  Astronomical  Journal,  October, 
1894,  he  called  specific  attention  to  irregularities  in  the 
motion  of  the  node  of  Venus,  and  in  the  motions  of  the 
node  and  eccentricity  of  Mercury.  The  final  results 
were  published  in  the  form  of  a  Supplement  to  the 
American  Ephemeris  and  Nautical  Almanac  for  1897, 
which  was  printed  in  Washington  in  1895.  Discrepan- 


174    Gravitation  versus  Relativity 

cies  between  the  calculated  and  observed  values  of  the 
secular  variations  were  found  to  exist  for  nearly  every 
element  of  the  four  planets;  four  of  these  discrepancies 
were  so  large  that  Newcomb  called  especial  attention  to 
them.  These  four,  as  enumerated  by  Newcomb,  are : 

1.  The  motion  of  the  perihelion  of  Mercury  +  41  ".6 

2.  The  motion  of  the  node  of  Venus  +  io".2 

3.  The  perihelion  of  Mars  +     8".i 

4.  The  eccentricity  of  Mercury  —    o".88 

In  this  final  table  of  results,  the  value  of  the  dis- 
cordance for  the  perihelion  of  Mercury  is  reduced 
somewhat  from  that  given  by  Newcomb  in  his  pre- 
liminary work,  heretofore  mentioned.  The  final  and 
definitive  value  is : 

+  4i"-6 

per  century.    In  the  same  table,  the  discordance  in  the 
eccentricity  of  Mercury  is  given  as : 
—  o".88 

In  his  entire  investigation  there  is  no  mention  of  the 
peculiar  connection  between  the  variations  of  the  peri- 
helion and  the  eccentricity,  so  clearly  brought  out  by 
Leverrier;  at  least,  Newcomb  does  not  directly  use  any 
such  equation  as  that  deduced  by  Leverrier  from  a  con- 
sideration of  the  May  and  November  transits.  Yet  the 
above  results  of  Newcomb,  derived  independently,  al- 
most exactly  satisfy  the  Leverrier  equation : 

2.72  e'  +  ;r'  = 


The  Perihelion  of  Mercury       175 

Thus  the  two  investigations  clearly  indicate  that  the 
motion  of  Mercury  cannot  be  explained  by  a  correction 
to  the  perihelion  alone;  there  must  be  a  corresponding 
correction  to  the  eccentricity.  Owing  to  the  greater 
magnitude  of  the  perihelial  discrepancy,  it  has  been 
stressed  in  all  discussions  and  has  been  given  a  promi- 
nent place  in  all  text-books,  while  the  corresponding, 
but  equally  important,  discrepancy  in  the  eccentricity 
has  been  completely  lost  sight  of,  and  no  mention  of 
it  is  to  be  found  in  any  text-book  or  popular  treatise 
on  the  motions  of  the  planets. 

While  these  two  magnificent  investigations  of  Lever- 
rier  and  Newcomb  place  the  existence  of  the  discrep- 
ancy in  the  motion  of  the  perihelion  beyond  all  doubt, 
they  do  not  definitely  fix  its  numerical  value.  The  re- 
sult of  Leverrier  differs  from  the  preliminary  result  of 
Newcomb  by  5",  and  is  3  ".6  smaller  than  Newcomb's 
final  definitive  value.  The  exact  numerical  value,  as- 
signed to  this  discrepancy,  is  a  compromise,  an  ad- 
justment of  many  individually  varying  observations. 
The  individual  observations  of  the  transits  vary  among 
themselves,  and  it  is  impossible  to  fix,  beyond  all  doubt, 
the  exact  instant  at  which  any  particular  phase  of  a 
transit  occurred.  Newcomb  and  Leverrier  differ  by 
six  seconds  in  their  adopted  times  for  a  single  phase 
of  the  transit  of  1845.  Newcomb  assigns  a  probable 
error  of  ±  i".4  to  his  final  result;  a  mathematical  way 
of  stating  that  the  chances  are  even  that  the  actual 


176    Gravitation  versus  Relativity 

error  in  his  determination  is  greater  or  less  than  this 
amount.  Taking  into  consideration  the  intricacies  of 
the  problem,  the  discordances  among  the  individual 
observations  of  the  recorded  transits,  and  the  intimate 
connection  between  the  motions  of  the  perihelion  and 
the  eccentricity,  it  is  perfectly  clear  that  the  actual 
value  of  the  discordance  in  the  motion  of  the  perihelion 
may  differ  by  some  seconds  from  that  given  by  New- 
comb.  The  mean  between  the  values  as  found  by 
Lever rier  and  Newcomb  is  39". 8,  practically  40".  And 
this  figure  probably  represents,  as  nearly  as  it  is 
humanly  possible  to  determine,  the  actual  discrepancy 
in  the  motion  of  the  perihelion. 

Since  the  "Vulcan"  fiasco  there  have  been  many 
attempts  to  find  an  explanation  of  the  excess  motion 
of  the  perihelion;  to  find  a  physical  cause  for  this 
motion.  Newcomb,  in  the  paper  above  mentioned, 
made  a  detailed  examination  of  all  the  suggestions, 
and  clearly  showed  that  the  difficulty  is  not  how  to 
account  for  the  motion  of  the  perihelion,  but  how  to 
account  for  that  motion  without  introducing  other 
complications  in  the  motions  of  Mercury  and  of  the 
other  planets.  He  showed  that  the  motion  of  the  peri- 
helion can  be  fully  accounted  for  by  any  one  of  several 
possible  distributions  of  matter  in  the  immediate  vicin- 
ity of  the  sun  and  the  inner  plants.  He,  however, 
discarded  each  possible  explanation  because  of  the  dif- 
ficulties encountered  in  explaining,  at  the  same  time,  the 


The  Perihelion  of  Mercury       17? 

motions  of  the  other  planets.  Each  possible  explana- 
tion of  the  motion  of  the  perihelion  introduced  a  new 
complication  somewhere  else  in  the  solar  system. 

The  simplest  explanation  is  to  be  found  in  the  first 
fundamental  approximation  of  celestial  mechanics.  It 
will  be  remembered  that  in  all  the  formulas  of  motion, 
in  all  the  numerical  calculations  of  the  perturbations, 
the  sun  and  all  the  planets  are  assumed  to  be  spheres  of 
uniform  density.  This  assumption  is  obviously  not 
true  for  the  planets,  and  is  probably  not  true  for  the 
sun.  A  simple  calculation  will  show  that  a  very  small 
departure  from  sphericity  in  the  sun  will  suffice  to  ac- 
count for  the  motion  of  the  perihelion  of  Mercury.  In 
fact,  if  the  equatorial  diameter  of  the  sun  should  be 
only  o".6  greater  than  the  polar,  then,  this  ellipticity 
alone  is  sufficient  to  give  the  perihelion  of  Mercury 
a  motion  of  40 "  per  century,  and  to  fully  reconcile  the 
difference  between  the  observed  and  the  calculated  mo- 
tions. Newcomb,  however,  pointed  out  two  difficulties 
in  the  way  of  accepting  this  explanation ;  the  heliometer 
measures  of  the  sun  made  by  the  German  observers, 
and  the  fact  that  any  such  ellipticity  of  the  sun  would 
introduce  other  discordances  into  the  system. 

An  ellipticity  of  the  sun  sufficient  to  produce  the 
observed  motion  in  the  perihelion  of  Mercury  would 
also  produce  various  motions  in  the  other  elements. 
It  would  give  rise  to  a  forward  motion  in  the  node 
of  Venus  and  thus  go  far  towards  accounting  for  the 


Gravitation  versus  Relativity 

second  large  discrepancy  pointed  out  by  Newcomb. 
But  it  would  also  cause  a  retrograde  motion  of  2". 6 
in  the  inclination  of  Mercury's  orbit.  The  motion  of 
the  inclination,  as  calculated,  is  smaller  than  the  ob- 
served motion  by  some  0^.38,  and  the  ellipticity  of  the 
sun  makes  this  discrepancy  worse,  not  better;  makes 
it,  in  fact,  practically  3".o.  The  whole  secular  motion 
of  the  inclination,  as  observed,  is  only  7".  14,  so  that 
a  discordance  of  3".o  is  42%  of  the  whole;  and  any 
such  disagreement  between  theory  and  observation  is, 
of  course,  wholly  out  of  the  question.  For  this 
reason,  Newcomb  justly  concludes  that  the  motion 
of  the  perihelion  cannot  be  explained  in  this  simple 
manner. 

On  the  other  hand,  it  is  practically  certain,  as  was 
pointed  out  in  a  former  chapter,  that  the  sun  is  slightly 
elliptical,  and  such  ellipticity  must  produce  some  motion 
in  the  perihelion.  This  ellipticity  is  probably  not  over 
o".io,  but  is  almost  certainly  at  least  one-half  of  this. 
For  the  sun  is  a  rotating  mass  of  gas,  and  the  normal 
shape  of  such  a  rotating  body  is  that  of  an  oblate 
spheroid.  From  the  known  dimensions  of  the  sun 
and  its  period  of  rotation  it  can  readily  be  shown 
that,  due  to  its  rotation,  the  equatorial  diameter  should 
exceed  the  polar  by  about  0^.05.  But  an  ellipticity 
of  even  this  amount  would  cause  a  motion  of  about 
3". 5  per  century  in  the  perihelion  of  Mercury;  an 
ellipticity  of  o".io,  the  amount  indicated  by  the  meas- 


The  Perihelion  of  Mercury       179 

urements  of  the  solar  disc,  would  give  rise  to  a  motion 
of  7".o  per  century. 

Some  portion  of  the  observed  discrepancy  is  thus 
undoubtedly  due  to  the  shape  of  the  sun;  but  this 
portion  is  relatively  small  and  cannot  amount  to  more 
than  one-sixth  of  the  total  necessary  for  a  complete 
explanation. 

Another  explanation  of  the  discrepancies  between 
theory  and  observation  is  to  be  found  in  the  second 
fundamental  approximation  of  celestial  mechanics, — 
the  assumption  of  empty  space.  It  is  known  that  this 
assumption  is  not  true,  that,  on  the  contrary,  there  are 
vast  quantities  of  bodies  occupying  the  spaces  between 
the  planets.  Leverrier,  who  first  called  attention  to 
the  discordances,  believed  that  a  full  explanation  of 
them  could  be  found  in  either  a  single  planet,  or  in  a 
group  of  planetoids  in  the  space  between  Mercury  and 
the  sun.  Newcomb  investigated  the  probable  action 
of  groups  of  planetoids,  or  rings  of  matter,  lying  in 
various  positions  between  the  sun  and  Mercury,  and 
between  Mercury  and  Venus,  and  of  an  extended  mass 
of  diffused  matter  similar  to  that  which  causes  the 
zodiacal  light.  Any  one  of  these  hypotheses  can  be 
used  to  fully  explain  the  discordance  in  the  motion  of 
Mercury's  perihelion,  but  each  and  every  one  is  dis- 
carded by  Newcomb  on  account  of  other  difficulties 
encountered.  According  to  him,  the  hypothesis  which 
best  represents  all  the  conflicting  motions  of  Mercury 


i8o    Gravitation  versus  Relativity 

and  Venus  is  that  of  a  ring  of  planetoids  between  the 
orbits  of  the  two  planets.  In  this  case  the  difficulty 
encountered  is  the  large  inclination  of  the  ring,  neces- 
sary to  account  for  the  motion  of  the  node :  Newcomb 
finds  this  inclination  to  be  7°. 5,  slightly  greater,  there- 
fore, than  the  inclination  of  Mercury's  orbit.  New- 
comb  deems  such  a  great  inclination  highly  improbable, 
believing  that  there  would  be  a  tendency  for  the  planes 
of  the  orbits  of  such  a  ring  of  planetoids  to  scatter 
themselves  midway  between  the  planes  of  the  larger 
planets  and  the  invariable  plane  of  the  solar  system. 
Regarding  the  possibility  of  orbits  of  the  necessary 
great  inclinations,  Newcomb  says, 

"In  admitting  such  orbits  we  encounter  difficulties 
which,  if  not  absolutely  insurmountable,  yet  tell  against 
the  probability  of  tjie  hypothesis." 

Yet  another  objection  raised  by  Newcomb  is  in 
connection  with  the  discordance  in  the  motion  of  the 
perihelion  of  Mars,  which  is  entirely  similar  to  that 
of  Mercury  and  amounts  to  8".i  per  century.  Those 
distributions  of  matter,  which  might  account  for  the 
motions  of  Mercury  and  Venus,  account  for  only  a 
very  small  portion  of  this  excess  motion  of  Mars; 
leaving  this  to  be  accounted  for  in  other  ways.  Rea- 
soning by  analogy,  if  the  motion  of  Mercury  is  due  to 
a  ring  of  undiscovered  matter,  then  that  of  Mars  ought 
most  certainly  to  be  due  to  the  known  ring  of  planet- 
oids, which  revolve  within  the  orbit  of  Jupiter.  Sev- 


The  Perihelion  of  Mercury       181 

eral  hundreds  of  these  bodies  are  known,  the  largest 
being  about  485  miles  in  diameter,  and  the  smallest 
probably  not  over  10  miles.  The  total  mass  of  the 
known  bodies  is  altogether  too  small  to  have  any 
appreciable  effect  upon  the  motions  of  Mars.  From 
the  number  and  magnitudes  of  those  already  known, 
Newcomb  tried  to  estimate  the  probable  mass  of  the 
entire  group.  Admitting  that  the  zodiacal  light  and  the 
"gegenschein"  are  probably  due  to  light  reflected  from 
these  bodies,  too  minute  to  be  seen  separately,  New- 
comb  concludes  that  the  total  mass  is  far  too  small  to 
produce  the  observed  effect.  He  regards  as  "unsatis- 
factory" the  hypothesis  that  the  motion  of  Mars  is 
due  to  the  presence  of  these  bodies. 

Newcomb's  conclusions  in  regard  to  the  existence 
of  a  group  of  planetoids,  sufficiently  large  to  account 
for  the  observed  discordances,  is  expressed  as  follows : 

"It  seems  to  me  that  the  introduction  of  the  action  of 
such  a  group  into  astronomical  tables  would  not  be  justi- 
fiable. The  more  I  have  reflected  upon  the  subject  the 
more  strongly  seems  to  me  the  evidence  that  no  such  group 
can  exist,  that  whatever  anomalies  exist  can  not  be  due  to 
the  action  of  unknown  masses  of  matter. 

"Besides,  the  six  elements  of  such  a  group  would  con- 
stitute a  complication  in  the  tabular  theory." 

Newcomb  was  confronted  by  the  necessity  of  having 
to  prepare  planetary  tables  for  the  use  of  the  American 
Ephemeris  and  Nautical  Almanac;  tables  which  would 


1 82    Gravitation  versus  Relativity 

best  represent  the  actual  motions  of  the  planets.  In 
order  to  prepare  such  tables,  a  reconciliation  of  some 
sort,  between  theory  and  observation,  had  to  be 
adopted.  He  found  that  practically  all  the  discrepan- 
cies, except  those  of  the  various  perihelia,  could  be 
reasonably  well  accounted  for  by  small  corrections 
to  the  assigned  masses  and  elements  of  the  planets,  and 
that  the  discordances  in  the  motions  of  the  perihelia  of 
Mercury  and  of  Mars  were  fairly  well  represented  by 
Hall's  hypothesis  regarding  the  law  of  gravitation. 
This  hypothesis  was  suggested  by  Hall  in  1894,  and  is 
that  the  gravitation  of  the  sun  is  not  exactly  as  the  in- 
verse square,  but  that  the  exponent  is  a  fraction  greater 
than  2  by  a  certain  minute  constant.  This,  of  course, 
is  a  direct  modification  of  Newton's  law. 

Under  Newton's  law  of  gravitation,  the  force  be- 
tween the  two  bodies  of  masses  m  and  m',  and  at  a 
distance  apart  equal  to  r,  is  given  by : 

-r,  mm' 

Force  =  — — 

Under  the  law  suggested  by  Hall,  the  exponent  2 
would  be  increased  by  a  very  small  fraction,  so  that 
the  force  between  the  two  bodies  would  be : 

m  m' 
Force  -  -^ 

When  8  is  a  very  minute  fraction,  this  law  results 
in  giving  the  perihelion  of  the  planet  a  small  forward 


The  Perihelion  of  Mercury       183 

rotation,  without  affecting  any  other  element.  In  this 
regard,  therefore,  this  Hall  hypothesis  is  strikingly 
similar  to  the  Einstein  law,  and  antedates  this  latter 
by  many  years.  Hall  found  that  when  8,  in  this 
formula,  was  made  equal  to 

o.oooooo  1 6 

then  the  forward  motion  of  the  perihelion  of  Mercury 
would  be  43"  per  century,  and  the  discrepancy  would 
be  fully  accounted  for. 

Such  a  modification  of  the  law  of  gravitation  would, 
of  course  affect  the  perihelia  of  all  the  other  planets, 
and  Newcomb  computed  these  as : 

For  Mercury  +  43 ".3 7 

Venus  +  1 6  .98 

Earth  +  10  .45 

Mars  +  5  .55 

These  quantities,  for  Mercury  and  for  Mars,  agree 
very  well  with  the  values  of  the  observed  discrepancies ; 
but  the  agreement  fails  utterly  in  the  cases  of  Venus 
and  the  earth. 

Now  what  Newcomb  actually  did  in  forming  his 
tables  was  to  compromise;  to  accept  no  theory  as  to 
the  actual  cause,  or  causes,  of  the  observed  irregulari- 
ties, but  to  adopt  corrections  to  the  elements  and 
masses  in  such  forms  that  they  can  be  readily  distin- 
guished from  the  values  derived  from  purely  theoretical 
grounds.  To  quote : 


1 84    Gravitation  versus  Relativity 

"What  I  finally  decided  on  doing  was  to  increase  the 
theoretical  motion  of  each  perihelion  by  the  same  -fraction 
of  the  mean  motion,  a  course  which  ivill  represent  the 
observations  without  committing  us  to  any  hypothesis  as 
to  the  cause  of  the  excess  of  motion,  though  it  accords 
with  the  result  of  HALL'S  hypothesis  of  the  law  of 
gravitation;  to  reject  entirely  the  hypothesis  of  the  action 
of  unknown  masses,  and  to  adopt  for  the  elements  what 
we  might  call  compromise  -values  between  those  reached  by 
the  preceding  adjustment  and  those  which  would  exist  if 
there  is  abnormal  action"  * 

*  The  Elements  of  the  Four  Inner  Planets  and  the  Fundamental 
Constants  of  Astronomy.  Washington,  1895. 


CHAPTER  VI 

THE  MOTIONS  OF  THE  PLANETS  AND 
THE  RELATIVITY  THEORY 

THE  EXCESS  MOTION  of  the  perihelion  of  Mercury 
furnishes,  according  to  Einstein,  a  complete  con- 
firmation of  the  Relativity  Theory.  In  attempting 
to  establish  his  theories  by  this  excess  motion,  Ein- 
stein constantly  refers  to  it  as  something  unique,  as 
the  sole  irregularity  in  the  solar  system.  This  idea 
that  Mercury  is  the  one  planet  to  exhibit  an  irregular- 
ity of  motion,  is  the  sole  exception  in  an  otherwise 
orderly  solar  system,  is  expressed  by  Einstein  in  clear, 
unequivocal  language.  There  can  be  no  misunder- 
standing of  phrases  such  as  the  following: 

"The  sole  exception  is  Mercury,  the  planet  which  lies 
nearest  the  sun" 

"That  for  all  the  planets,  with  the  exception  of  Mercury, 
this  rotation  is  too  small  to  be  detected  with  the  delicacy 
of  the  observation  possible  at  the  present  time." 

The  PREFACE  to  the  book  "Relativity,"  in  which 
these  statements  appear,  was  written  in  December, 
1916:  the  third  edition  was  published  in  1920.  And 

185 


i86    Gravitation  versus  Relativity 

this  definite  position  of  the  author  of  Relativity  in 
regard  to  the  motion  of  Mercury  is  still  more  clearly 
established  by  a  letter  written  by  him  on  July  30,  1921 
— a  letter  written  for  publication  and  which  has  been 
printed.  In  this  letter  appears  the  clear  cut  declaration : 

"The  perihelial  movement  of  Mercury  is  the  only 
anomalous  one  in  our  planetary  system  which  has  been 
sufficiently  attested." 

Yet  Leverrier  in  1859,  in  the  very  paper  in  which  he 
announced  the  discovery  of  this  "perihelial  movement," 
showed  that  its  very  existence  connotes  a  similar 
irregularity  in  the  eccentricity;  Newcomb  in  1894 
specifically  called  attention  to  discordances  in  the  mo- 
tions of  other  planets,  and,  in  his  final,  definitive  work 
published  in  1895,  ne  gives  the  amounts  of  eleven 
discordances  and  calls  particular  and  especial  attention 
to  four  of  these.  Thus,  according  to  Newcomb,  there 
are,  at  least,  four  anomalous  motions  in  the  planetary 
system,  which  have  been  fully  attested;  and  his  state- 
ment is  supported  by  a  classic  research  involving  over 
60,000  observations  and  by  a  life-time  devoted  to  the 
study  of  similar  questions.  Against  these  statements 
of  Leverrier  and  Newcomb,  statements  backed  up  by 
years  of  patient  astronomical  research,  Einstein  sets 
his  mere  opinion,  the  opinion  of  one,  who,  however 
eminent  he  may  be  as  a  mathematician,  is  not  an 
astronomer,  and  has  never  made,  so  far  as  known,  an 


The  Motions  of  the  Planets      187 

astronomical  investigation  of  this  character.  Can  it 
be  that  the  author  of  Relativity  is  unaware  of  these 
statements  of  Newcomb  ?  Can  it  be  possible  that  he  has 
never  read  the  very  papers,  upon  which  the  astronomi- 
cal proof  of  the  Relativity  Theory  is  supposed  to  be 
based? 

The  Relativity  Theory  can  explain  a  certain  definite 
amount  of  motion  in  the  perihelion  of  Mercury.  Ac- 
cording to  the  formulas  and  computations  of  Einstein 
this  amounts  to  43"  per  century;  it  can  account  for 
neither  more  nor  less.  There  is  no  flexibility  in  the 
Einstein  formulas,  no  constant  of  uncertain  value,  no 
possibility  of  adjustment.  The  difference  between 
the  actual  motion  of  the  perihelion  and  the  theoretical 
motion  under  the  Newtonian  law  of  gravitation  must 
be  exactly  this  amount,  if  the  Einstein  theories  are 
true :  no  other  amount  will  satisfy  the  principles  of 
relativity.  The  actual  motion  can  be  determined  by 
observation  only;  the  theoretical  motion  by  calculation 
alone;  and  it  is  certainly  remarkable  that  the  difference 
between  observation  and  theory  should  be  exactly  the 
43"  required  by  the  relativity  theory.  This  is  a  most 
striking  coincidence,  and  this  coincidence  has  been 
stressed  as  proof  positive  and  conclusive  of  the  Ein- 
stein doctrines.  Yet  this  coincidence  of  figures  is 
largely  due  to  the  astuteness  of  Einstein  in  quoting  the 
result  of  Newcomb's  preliminary  investigation,  and  in 
ignoring  the  classic  work  of  Leverrier  and  the  final 


1 88    Gravitation  versus  Relativity 

results  of  Newcomb.  According  to  Einstein  the  re- 
sults of  the  astronomical  investigations  into  the  motions 
of  Mercury  are  summed  up  as : 

"it  was  found  (Leverrier — 1859 — and  Newcomb — 1895) 
that  an  unexplained  perihelia!  movement  of  the  orbit  of 
Mercury  remained  over,  the  amount  of  which  does  not 
differ  sensibly  from  the  above  mentioned  +  43  seconds 
of  arc  per  century.  The  uncertainty  of  the  empirical 
result  amounts  to  a  few  seconds  only"  (152). 

Leverrier  in  1859  found  38" :  Newcomb  in  1895 
found  4 1 ".6;  quantities  quite  different  from  the  43" 
quoted  by  Einstein.  This  latter  figure  was  the  result 
of  Newcomb's  first  investigation,  published  in  1882; 
it  does  not  appear  in  Newcomb's  final  work  as  pub- 
lished in  1895.  In  one  portion  of  this  classic  investi- 
gation Newcomb  makes  use  of  a  "provisional  value  of 
40". 7."  His  definitive  value  is  found  only  in  the 
tables,  where  it  appears  as  multiplied  by  the  eccen- 
tricity. He  gives  for  this  product  the  value  +  8". 48; 
from  which  by  simple  division  it  is  easy  to  derive  the 
specific  amount  +  41  ".6,  as  the  excess  motion  of 
the  perihelion.  The  mean  of  the  two  results — Lever- 
rier 1859  and  Newcomb  1895 — is  39". 8,  which  differs 
by  3 ".2,  by  8  per  cent,  from  the  figure  quoted  by  Ein- 
stein. The  coincidence  of  figures,  the  supposed  agree- 
ment between  observation  and  the  relativity  theory, 
vanishes  the  moment  the  real  facts  are  stated. 

Further,  as  has  been  pointed  out,  the  sun  is  a  rotating 


The  Motions  of  the  Planets      189 

body;  and  from  its  known  size,  density,  and  period  of 
revolution  can  be  computed  its  theoretical  shape  as  an 
oblate  spheroid.  And  such  computations  rest  upon 
physical  laws  and  can  be  made  with  the  same  theoreti- 
cal accuracy  that  Einstein  uses  in  finding  the  ' 'relativity 
motion"  of  Mercury.  Such  a  theoretical  oblateness  of 
the  sun  would  cause  a  motion  of  at  least  3 ".5  per 
century  in  Mercury's  perihelion.  The  actual  measure- 
ments of  the  sun  indicate  a  somewhat  greater  oblate- 
ness,  and  hence  a  larger  motion  in  the  perihelion.  But, 
using  the  theoretical  value  and  deducting  it  from  the 
observed  value  of  39". 8,  there  is  left  over  an  unex- 
plained motion  of  the  perihelion  of  Mercury  of  not 
over  36". 3.  Thus  the  real  amount  of  the  unexplained 
perihelial  motion  of  Mercury  differs  by  6".  7  from  the 
43"  of  the  Einstein  theory,  differs  by  16  per  cent  of 
the  required  amount.  And  such  a  difference  is  very 
nearly  fatal  to  the  Relativity  Theory,  for  that  theory 
contains  no  arbitrary  constant  by  which  Einstein 
can,  in  the  future,  readjust  his  figures  to  fit  the  real 
and  not  the  imaginary  facts. 

The  relativity  theory,  thus,  does  not  satisfactorily 
explain  even  the  one  discordance  in  planetary  motions, 
which  has  been  so  thoroughly  exploited  by  Einstein 
and  his  followers.  If  his  methods  and  formulas  fail 
to  meet  the  test  in  this  one  selected  case,  selected  by 
the  author  of  relativity,  how  do  these  same  formulas 
and  methods  fare  in  tests  with  other  observed  planetary 


190    Gravitation  versus  Relativity 

discordances?  For,  Einstein  to  the  contrary  notwith- 
standing, there  are  many  anomalous  movements  in 
the  planetary  system;  anomalies,  which  should  be 
explained  and  accounted  for  by  any  hypothesis  or 
theory  which  pretends  to  remake  the  universe.  In  his 
discussion  of  the  elements  of  the  four  inner  planets 
Newcomb  found  eleven  anomalies  among  the  fifteen 
secular  motions,  and  of  these  eleven  he  singled  out 
four  as  requiring  especial  consideration. 

Now,  the  formulas  and  theories  of  Einstein  differ 
from  the  formulas  of  the  Newtonian  mechanics  in  one 
point  and  one  point  only.  Under  the  relativity  formu- 
las the  perihelion  of  a  planet  will  have  a  forward  rotary 
motion;  but  every  other  element  and  motion  of  the 
planet  will  be  the  same  as  that  under  the  Newtonian 
law.  Mathematical  discussions  of  the  Einstein  theory, 
too  formidable  to  enter  into,  show  that,  "The  only 
secular  perturbation  is  a  motion  of  the  perihelion."* 
Thus  the  relativity  theory  cannot  explain,  or  account 
for,  any  of  the  observed  discrepancies  in  the  motions 
of  the  planets,  other  than  those  in  the  perihelia. 

But  it  is  clear  that,  under  the  relativity  theory,  the 
perihelia  of  all  the  planets  must  rotate  by  various 
amounts  depending  upon  their  respective  distances 
from  the  sun.  The  amounts  of  such  rotations  can  be 

*  On  Einstein's   Theory   of  Gravitation,  and  its  Astronomical 
Consequences,  by  W.  de  Sitter. 
Monthly  Notices,  R.A.S.,  vol.  Ixxvi,  No.  9,  page  726. 


The  Motions  of  the  Planets      191 

readily  calculated  from  the  formula  given  by  Einstein 
for  the  case  of  Mercury.  This  Einstein  rotation  is 
independent  of  the  mutual  action  of  the  planets  upon 
one  another  and  may  be  directly  compared  with  the 
discordances  as  determined  by  Newcomb.  The  results 
of  such  comparison  are  shown  in  the  following  table  : 

TABLE  II 

Observed  Discordances  and  the  Einstein  Motions 

Final 


-  is".9 
-f  2".! 
+  6".8 


io".2 

ECCENTRICITY: 

(4)  Mercury   -    o".88  ±  0^.50  26.5%  O        -    o".88 

The  first  column,  in  this  table,  gives  the  values  of 
the  discordances  between  theory  and  observation  as 
determined  by  Newcomb  :  the  large  bracketed  numbers 
give  the  order  in  which  Newcomb  arranged  the  four 
especially  large  discrepancies.  Each  one  of  these  dis- 
cordances is  followed  by  the  so-called  "probable  error" 
of  the  determination.  These  probable  errors  give  some 
idea  as  to  the  relative  accuracy  of  the  various  deter- 


PERIHELIA: 

Tabular 
Discordances 

Per 

cent 

Einstein 
motion 

(i) 

Mercury 

+ 

41"- 

,6  d 

=     i"4 

7.2% 

4-42 

"•9 

Venus 
Earth 

4- 

£ 

•3  d 
•9  d 

=   22".3 

=    5"-6 

17-2% 
0.5% 

+ 
+ 

8 
3 

".6 

".8 

(3) 

Mars 

+ 

8" 

.1  d 

b     2".6 

0.5% 

+ 

i 

"•3 

NODES: 

Mercury 

+ 

5"  I  =t 

:    2".8 

0.7% 

0 

(2) 

Venus 

+ 

10". 

2  d: 

:     2".0 

0.6% 

o 

i92    Gravitation  versus  Relativity 

minations,  but  it  must  be  remembered  that  the  assign- 
ment of  these  probable  errors  is  very  largely  a  matter 
of  judgment,  and  that  these  values  may  be  over-  or 
under-estimated.  In  every  step  of  the  long  and  com- 
plicated computations  an  estimate,  rather  than  an 
exact  calculation,  has  to  be  made  as  to  the  value  of 
the  probable  error,  and  the  final  value,  as  given  in  the 
table,  thus  depends  upon  many  separate  estimations  or 
judgments. 

The  second  column  gives  the  percentage  that  the 
discordance  bears  to  the  observed  motion.  While  this 
is  rather  an  unusual  way  of  comparing  results,  it  may- 
be of  interest  and  it  may  throw  some  light  upon  the 
problem.  It  will  be  noted  that  three  of  the  discordances 
represent  very  large  percentages  of  the  observed 
quantities ;  and  of  these  three,  two  are  among  the  dis- 
cordances specifically  singled  out  by  Newcomb.  Of 
the  remaining  two,  that  of  the  perihelion  of  Venus  is 
peculiar.  In  the  case  of  this  planet  the  computed  value 
of  the  motion  of  the  perihelion  is  greater  than  the  ob- 
served motion,  thus  giving  the  discordance  the  negative 
sign.  Again  this  discordance  is  over  17  per  cent  of 
the  total  observed  motion,  yet  Newcomb  estimates  it 
as  being  only  about  one-third  of  the  probable  error. 
It  would  seem  that  where  the  percentage  is  so  large, 
the  discordance  must  be  real  and  that  the  size  of  the 
probable  error  has  been  over-estimated. 

The  third  column  gives  the  Einstein  motion,  as  com- 


The  Motions  of  the  Planets      193 

puted  from  his  formulas,  and  the  final  column,  the 
discrepancies  between  theory  and  observation  after 
allowing  the  full  and  complete  Einstein  effect. 

An  inspection  of  this  table  shows  that  the  Einstein 
motion  is  sufficient  to  account  for  practically  all  the 
"tabular"  discrepancy  in  the  motion  of  the  perihelion 
of  Mercury,  and  to  reduce,  in  a  marked  manner,  that 
for  the  earth.  It  accounts,  however,  for  only  a  very 
small  portion  of  the  discordance  in  the  case  of  Mars, 
and  more  than  doubles  the  already  large  discrepancy 
in  the  case  of  Venus.  It  does  not  account  for  the 
motion  of  Mars  nearly  so  well  as  does  the  Hall  hypo- 
thesis. And  further,  the  Einstein  law  does  not  in  any 
way  account  for  the  important  discrepancies  in  the 
motions  of  the  nodes  and  in  the  eccentricity  of  Mercury. 

Einstein,  himself,  disregards  the  motions  of  all  the 
planets  except  Mercury,  dismissing  from  consideration 
their  perihelial  movements  with  the  simple  statement, 
"for  the  other  planets  of  our  solar  system  its  magni- 
tude should  be  so  small  that  it  would  necessarily  escape 
detection/'  This  statement  may  be  perfectly  true  for 
some  of  the  theoretical  Einstein  motions,  which  in 
the  case  of  Mars  amounts  to  only  i".3,  but  it  certainly 
does  not  apply  to  the  observed  motions  of  the  perihelia, 
and  it  conveys  a  distinctly  erroneous  impression. 
Twenty-five  years  before  Einstein  wrote  these  words, 
Newcomb  detected  and  measured  such  movements, 
found  that  the  perihelion  of  Mars  rotates  by  an  un- 
13 


*94    Gravitation  versus  Relativity 

explained  8".i  per  century,  by  an  amount  six  times 
the  whole  Einstein  effect.  It  is  hard  to  see  upon 
what  scientific  grounds  it  is  allowable  to  select  one 
result  of  a  scientfic  research  and  to  dismiss  all  the 
others  as  negligible,  why  one  figure  is  to  be  taken  as 
absolutely  accurate  and  all  other  figures  thrown 
out  as  worthless !  It  certainly  cannot  be  a  question  of 
mere  size,  that  43"  is  accurately  measurable,  and  that 
8"  is  too  small  even  to  be  detected.  In  many  an  astro- 
nomical research  8"  is  a  quantity  of  supreme  impor- 
tance. The  fundamental  unit  of  astronomy,  the  solar 
parallax,  is  only  8". 8;  the  parallaxes  of  the  fixed  stars 
are  measured  in  minute  fractions  of  a  single  second. 

The  motion  of  the  perihelion  of  Venus  is  peculiarly 
embarrassing  for  the  relativity  theory.  According  to 
Newcomb's  results,  the  perihelion  of  this  planet  is 
rotating  more  slowly  than  the  computations  indicate 
it  should,  the  difference  being  7". 3  per  century.  The 
Einstein  formulas  would  increase  the  theoretical  speed 
of  rotation  by  an  additional  8". 6,  thus  making  the  total 
discrepancy  between  observation  and  theory  15 ".9,  or 
37  per  cent  of  the  entire  observed  motion!  The  Ein- 
stein formulas,  in  this  case,  make  a  bad  matter  worse; 
they  give  the  orbit  a  rotation  in  the  opposite  direction 
to  that  which  is  required  to  fit  the  observations.  It  is 
perfectly  true  that  the  eccentricity  of  the  orbit  is  small 
and  that,  on  this  account  the  Newcomb  determination 
of  the  motion  is  liable  to  a  much  larger  error  than 


The  Motions  of  the  Planets      195 

in  the  cases  of  other  planets.  But  it  is  hardly  likely 
that  any  such  error  would  be  large  enough  to  change 
the  direction  of  motion. 

Thus  the  Relativity  Theory  is  not  sufficient  to  ex- 
plain the  discordances  in  the  planetary  motions.  It 
accounts  approximately  for  only  one  among  the  nu- 
merous discrepancies — that  of  the  perihelion  of 
Mercury.  It  fails  completely  to  explain  any  portion 
of  several  well-attested  irregularities — those  of  the 
nodes  and  eccentricities;  and  it  doubles  the  observed 
discrepancy  in  the  motion  of  Venus.  A  simple  in- 
vestigation will  show  that  the  theory  is  not  necessary 
to  explain  even  the  one  discordance  which  it  can  more 
or  less  account  for: — the  motion  of  the  perihelion  of 
Mercury  can  be  fully  accounted  for  by  the  actibn, 
under  the  Newtonian  law,  of  matter  known  to  be  in 
the  immediate  vicinity  of  the  sun  and  the  planets. 

Newcomb,  many  years  ago,  showed  that  this  motion 
can  be  completely  accounted  for  by  any  one  of  several 
possible  distributions  of  matter  in  or  near  the  sun. 
The  difficulty  which  faced  Newcomb  and  the  astrono- 
mers of  the  Newtonian  school,  is  not  how  to  account 
for  the  motion  of  Mercury,  but  how  to  account  for  it 
in  such  a  way  as  to  explain,  at  the  same  time,  the  other 
observed  irregularities.  This  difficulty,  which  appeared 
nearly  unsurmountable  to  Newcomb,  is  readily  dis- 
posed of  by  Einstein  by  the  simple  expedient  of  saying 
that  such  other  irregularities  do  not  exist,  or  rather 


196    Gravitation  versus  Relativity 

have  not  "been  sufficiently  attested."  If  this  "rela- 
tivity" method  of  surmounting  the  difficulty  be  adopted 
by  the  astronomer,  then  is  he  faced  by  an  embarrass- 
ment of  riches  and  the  problem  is  reduced  to  one  of 
mere  choice,  to  the  selection  of  that  solution  which 
best  pleases  one's  fancy.  For  the  motion  of  the  peri- 
helion of  Mercury,  taken  by  itself,  can  be  explained 
equally  well  by: 

1.  A  non-spherical  sun. 

2.  A  ring  of  matter  between  Mercury  and  the  sun. 

3.  A  group  of  planetoids  outside  the  orbit  of  Mercury. 

4.  The  Hall  hypothesis. 

And  there  are  good  reasons  for  accepting  each  one  of 
the  three  first  mentioned :  the  sun  is  non-spherical,  and 
matter  is  known  to  exist  both  within  and  without  the 
orbit  of  Mercury. 

But,  if  the  methods  of  the  author  of  relativity  are 
to  be  admitted,  there  is  no  necessity  of  explaining  the 
perihelial  motion  of  Mercury.  If  it  is  troublesome  to 
our  theories,  it  can  be  discarded  along  with  all  the  other 
discordances.  Why  even  bother  about  Mercury  itself ! 
Copernicus  is  said  never  to  have  seen  the  planet;  and 
the  solar  system  would  really  be  much  simpler  with- 
out it ! 


CHAPTER  VII 

THE    ECLIPSE    PLATES    AND    THE    RELATIVITY    THEORY 

THE  SECOND  ASTRONOMICAL  proof  of  the  Relativity 
Theory,  cited  by  Einstein,  is  apparently  supported  by 
very  much  stronger  evidence.  This  proof,  as  shown 
in  Chapter  II,  rests  upon  the  deflection  of  light  ob- 
served by  the  British  astronomers  at  the  total  solar 
eclipse  of  May  29,  1919.  As  presented  in  the 
"Report,"  the  evidence  makes  a  strong  prima  facie 
case  for  the  Relativity  Theory.  It  was  this  evidence 
and  the  way  in  which  it  was  presented  to  the  Royal 
Astronomical  Society  at  the  meeting  of  November 
6,  1919,  that  caused  the  furor  in  regard  to  the  Ein- 
stein theory  and  its  acceptance  by  so  many  scientists. 
But  on  an  examination  it  will  be  seen  that  the  strength 
of  this  evidence  has  been  greatly  magnified,  and  that 
many  elements,  which  tend  to  weaken  its  force,  have 
been  omitted  from  the  public  announcements  and  from 
popular,  or  semi-scientific  expositions. 

The  eclipse  expedition  of  1919  was  organized  by 
the  leading  scientists  of  England,  and  was  partici- 
pated in  by  astronomers,  trained  through  long  years  of 
scientific  research.  The  final  report,  giving  the  results 

197 


198    Gravitation  versus  Relativity 

of  the  expedition,  was  drawn  up  by  Sir  F.  W.  Dyson, 
F.R.S.,  Astronomer  Royal,  Prof.  A.  S.  Eddington, 
F.R.S.,  and  Mr.  C.  Davidson.  This  report  is  a  scien- 
tific paper  of  the  highest  possible  value,  and  will,  un- 
doubtedly, rank  among  the  classic  papers  of  astronomy. 
In  it  are  to  be  found  the  scientific  details  of  the  expe- 
dition, presented  in  a  clear  and  well-ordered  manner. 
But,  as  the  report  was  prepared  by  eminent  astrono- 
mers, accustomed  to  the  intricacies  of  astronomical 
methods,  many  of  the  essential  details  are  embodied  in 
long  and  complicated  looking  tables,  which,  while 
perfectly  clear  to  the  reader  trained  in  these  methods, 
are  meaningless  to  the  average  scientist,  be  he  a  mathe- 
matician or  a  physicist.  Few,  if  any,  scientists,  other 
than  a  small  group  of  astronomers,  have  probably  ever 
read  anything  more  than  the  concluding  paragraphs 
of  the  Report. 

In  the  following  pages  an  attempt  is  made  to  present 
the  essential  elements  of  the  Report  and  the  results 
of  the  expedition  in  unteclmical  language. 

The  expedition  was  divided  into  two  parties;  one 
of  which,  under  the  direction  of  Professor  Eddington, 
went  to  the  island  of  Principe,  off  the  west  coast  of 
Africa;  and  the  second  party,  under  the  direction  of 
Dr.  Crommelin,  to  Sobral,  a  small  town  some  miles 
inland  from  the  north-east  coast  of  Brazil.  The  two 
parties  were  equipped  with  horizontal  photographic 
telescopes;  the  object  glasses  of  which  were  borrowed 


The  Eclipse  Plates  199 

from  Oxford  and  Greenwich  respectively.  These 
lenses,  known  as  astrographic,  were  made  especially 
for  photographic  work  and  are  13  inches  in  diameter. 
The  horizontal  type  of  mounting,  used  by  the  expe- 
dition, is  very  convenient  for  field  work,  and  has  been 
almost  exclusively  used  in  eclipse  expeditions  but  it 
involves  some  disadvantages  for  accurate  measure- 
ments, as  will  be  seen  by  reference  to  the  following 
diagram. 

The  telescope,  itself,  is  mounted  horizontally  in  a 
fixed  position,  the  tube  resting  upon  temporary  piers. 
In  front  of  the  telescope  lens,  B,  is  mounted  a  plane 
mirror,  A,  which  is  driven  by  clock-work ;  such  a  mov- 
ing mirror  being  known  as  a  ccelostat.  The  mirror  of 
such  instruments  are  made  of  heavy  glass  and  are 
coated  with  a  thin  film  of  silver  on  the  front  surface ; 
the  light  being  reflected  from  the  silver  face  and  not 
entering  the  glass  itself.  A  ray  of  light  from  the 
eclipsed  sun  is  reflected  from  the  mirror  into  the  lens 
of  the  horizontal  telescope,  where  it  passes  to  the 
photographic  plate  at  C.  Now,  as  the  sun  rises  in  the 
heavens  the  mirror  must  be  rotated,  as  otherwise 
the  reflected  beam  would  soon  pass  off  the  lens  of  the 
telescope,  and  such  rotation  is  provided  by  the  clock- 
work. As  the  mirror  is  generally  in  an  inclined 
position,  its  diameter  must  be  greater  than  that  of  the 
lens,  otherwise  the  beam  of  reflected  light  would  be 
too  narrow  to  cover  the  entire  surface  of  the  lens. 


200 


The  Eclipse  Plates  201 

The  mirrors,  used  by  the  expedition,  were  16  inches 
in  diameter. 

It  will  be  at  once  apparent  that  such  a  horizontal 
arrangement  is  much  more  convenient  than  the  ordi- 
nary type  of  telescope  mounting.  In  the  latter  the  tube 
of  the  telescope,  BC,  is  pointed  directly  at  the  sun,  or 
other  heavenly  body,  the  mirror  being,  of  course,  dis- 
pensed with.  But  the  tube,  in  this  case,  must  be  hung 
on  axes,  so  as  to  follow  the  movements  of  the  body 
as  it  travels  across  the  heavens.  Such  a  direct  telescope 
mounting,  especially  for  a  large  lens,  is  an  elaborate 
piece  of  mechanism,  involving  very  large  and  heavy 
parts.  And  such  mounting  must  be  constructed  for 
the  locality  in  which  it  is  to  be  used ;  so  that  the  heavy, 
permanent  mounting  constructed  for  Oxford  would 
be  of  no  use  in  Brazil,  or  on  the  island  of  Principe. 
It  would  be  practically  impossible  to  modify  such  a 
mounting  and  to  transport  it  to  some  out  of  the  way 
district,  and  there  erect  it  for  temporary  use.  On  the 
other  hand  the  astronomers  of  the  Lick  Observatory 
of  California  have  evolved  a  form  of  temporary 
mounting  suitable  for  field  work  in  which  the  mirror 
is  discarded  and  the  lens  is  always  pointed  directly  at 
the  eclipsed  sun. 

For  many  purposes  the  horizontal  telescope  is 
perfectly  satisfactory:  for  obtaining  photographs  of 
the  corona,  for  spectroscopic  investigations,  the  intro- 
duction of  the  mirror  is  no  disadvantage.  But  for 


202    Gravitation  versus  Relativity 

exact,  minute  measurements  the  mirror  introduces 
complications.  In  the  first  place  the  instrument  is  no 
longer  symmetrical;  horizontal  and  vertical  distances 
in  the  heavens  are  reflected  differently  by  the  mirror, 
and  this  asymmetry  will  always  introduce  an  element 
of  doubt  as  to  the  value  of  results.  Again,  a  mirror 
is  far  more  delicate  than  a  lens :  a  slight  distortion  in 
the  shape  of  a  mirror  will  produce  approximately  three 
times  the  effect  upon  the  image  that  a  similar  dis- 
tortion in  a  lens  will  produce.  Very  slight  changes  in 
temperature  will  produce  marked  distortions  in  the 
shape  of  a  mirror;  the  mere  tilting  of  a  mirror  from 
one  position  to  another,  causing  its  own  weight  to 
rest  more  on  one  portion  than  another,  will  often 
so  change  the  shape  of  a  mirror  as  to  cause  notice- 
able distortion  in  the  image.  These  facts  have 
been  known  for  many  years,  certainly  ever  since 
the  transit  of  Venus  in  1882,  when  the  results  ob- 
tained with  horizontal  telescopes  were  not  satisfactory. 
In  addition  to  the  regular  astrographic  object  glass, 
the  Sobral  party  took,  as  an  auxiliary,  a  4-inch  tele- 
scope of  1 9- feet  focus,  loaned  by  the  Royal  Irish 
Academy.  This  instrument  had  been  used  in  conjunc- 
tion with  an  8-inch  mirror  by  Father  Cortie  in  Sweden 
in  1914.  This  lens  was  mounted  in  a  square  wooden 
tube,  and  the  mounting  of  the  mirror  was  adjusted 
for  the  difference  of  latitude  between  England  and 
Brazil  by  bolting  it  upon  a  strong  wooden  wedge. 


The  Eclipse  Plates  203 

On  a  photographic  plate  taken  with  this  instrument 
a  second  of  arc  (i")  is  represented  by  a  distance  of 
approximately  one  nine-hundredth  (i/cjooth)  of  an 
inch. 

Now  before  discussing  in  detail  the  actual  results 
obtained  by  the  two  eclipse  parties,  it  will  be  well  to 
understand  something  of  the  methods  and  the  diffi- 
culties involved.  A  photograph  of  a  portion  of  the 
heavens  shows  the  various  stars  as  minute  dots,  the 
images  of  the  brighter  stars  being  larger  than  those  of 
their  fainter  companions.  The  principal  stars  of  the 
eclipse  field  and  their  relative  positions  are  shown  in 

S 

I 

• 

•  .5 


•\      ) 

-£• 
•3 


I 
N 


•6 

•10 


Fig.  25.    The  Eclipse  Field. 

the  accompanying  diagram,  in  which  the  dotted  circle 
represents  the  eclipsed  sun.  The  numbers,  near  each 
star,  are  those  used  in  the  Report  for  identification. 


204    Gravitation  versus  Relativity 

The  unnumbered  stars  did  not  show  on  the  photo- 
graphic plates,  or  not  on  a  sufficient  number  of  plates 
to  be  of  any  use. 

The  original  photographic  plate,  the  negative  itself, 
is  put  in  a  measuring  machine,  under  microscopes, 
and  the  relative  positions  of  the  images  carefully  deter- 
mined. The  measurements  are  made  with  a  carefully 
calibrated  micrometer  screw,  and  are  given  in  terms  of 
revolutions  of  the  screw.  These  screw  revolutions  must 
be  turned  into  seconds  of  arc  by  determining  the  scale 
value  for  the  particular  plate  or  plates  being  measured. 
This  scale  value  depends  upon  the  distance  of  the  plate 
from  the  optical  centre  of  the  lens,  upon  the  focal 
length  of  the  lens  in  other  words.  The  focus  of  a  lens 
does  not  remain  absolutely  constant,  especially  when 
the  lens  is  used  under  widely  varying  conditions  of 
climate  and  temperature.  Therefore  the  scale  of 
photographs  taken  with  the  same  lens  at  different 
times  and  places  may  vary.  Again  it  is  impossible  to 
exactly  orient  a  plate  and  this  introduces  another  diffi- 
culty. But  standard  methods  of  measurement  and 
reduction  have  been  worked  out  by  which  these  and 
other  instrumental  difficulties  are  overcome,  and  the 
results  freed  from  their  effects. 

Now  the  distance  between  the  various  stars  on  the 
eclipse  plates  are  large  and  the  direct  measurement  of 
these  distances  would  be  difficult,  so  an  indirect  method 
was  adopted.  A  special  photograph  of  the  region 


The  Eclipse  Plates  205 

was  taken  with  the  plate  reversed,  so  that  the  light 
passed  through  the  glass  plate  before  reaching  the 
film.  When  this  plate  was  developed,  it  could  be 
placed  in  contact  with  one  of  the  regular  eclipse  plates, 
film  to  film,  and  the  images  of  the  same  star  on  the 
two  plates  would  appear  very  close  together.  The 
small  distances  between  the  corresponding  images  of 
the  two  plates  could  be  measured  with  extreme  ac- 
curacy. This  special,  or  scale  plate,  merely  provided 
a  convenient  point  of  reference  near  each  star,  to 
which  could  be  referred  the  images  on  each  and  all 
the  plates  in  turn.  While  this  method  gives  with  ex- 
treme accuracy  the  shift  of  the  images  from  plate  to 
plate,  it  does  not,  of  course,  give  the  actual  coordinates 
of  the  various  stars.  But  it  gives  accurately  and 
quickly  everything  that  is  necessary  for  the  investiga- 
ion  of  possible  light  deflections. 

There  is  in  the  whole  method,  however,  an  inherent 
astronomical  difficulty,  a  difficulty  that  is  really  seri- 
ous. The  light  from  the  stars  passes  through  the 
atmosphere  of  the  earth,  and  is  refracted,  or  bent  out 
of  its  course.  Each  star  appears  higher  in  the  heavens 
than  it  really  is,  and  the  amount  of  this  refractive 
effect  depends  upon  the  height  of  the  star  above  the 
horizon,  upon  the  temperature  of  the  air,  and  upon 
the  barometric  pressure.  At  any  given  instant,  such 
as  that  at  which  a  photograph  is  taken,  the  various 
stars  on  the  plate  will  be  at  different  heights  above 


2o6    Gravitation  versus  Relativity 

the  horizon,  and  will,  therefore,  be  refracted  differently. 
It  is  this  "differential  refraction"  as  it  is  called,  that 
is  of  importance,  for  in  the  present  investigation  one 
is  interested  solely  in  the  relative  positions  of  the 
various  stars,  and  not  at  all  in  the  actual  height  of 
the  group,  as  a  whole,  above  the  horizon.  The  amount 
of  this  differential  refraction  is  calculated  by  theoreti- 
cal formulas,  derived  for  standard  conditions  of  the 
atmosphere,  with  corrections  for  temperature  and 
pressure.  The  entire  amount  of  the  refraction  for 
the  stars  on  the  eclipse  plates  was  approximately  one 
hundred  times  the  sought-for  shift,  and  the  differential 
refraction  between  the  various  stars  amounted  to 
several  times  the  full  expected  light  deflections. 

Further  the  actual  amount  of  the  refraction  at 
any  time  depends  largely  upon  the  rate  of  decrease 
of  temperature  of  the  air  with  the  height  above  the 
surface  of  the  earth,  and  this  rate  may  be  quite  differ- 
ent in  day-time  from  night-time.  For  the  sun  heats 
the  surface  of  the  earth,  and  the  air  near  the  surface 
is,  therefore,  warmer  in  day-time  than  at  night.  As 
most  astronomical  observations  are  made  at  night,  the 
tables  of  refraction  represent  night  observations  and 
night  conditions  of  the  atmosphere,  rather  than  day 
observations  and  day  conditions.  For  this  reason, 
there  is  a  certain  element  of  uncertainty  introduced, 
when  the  eclipse  plates  are  corrected  for  the  effects 
of  refraction  by  the  use  of  tables  primarily  calculated 


The  Eclipse  Plates  207 

for  and  based  upon  night  observations.     Again,  cases 
of  abnormal  refraction  are  not  infrequent;  instances 
where  local  currents  of  hot  or  cold  air  completely 
change  the  refraction  from  that  given  in  the  standard 
tables.    During  an  eclipse  the  heat  of  the  sun  is  with- 
drawn from  a  funnel  shaped  column  of  air  over  the 
observing   station,   and   this   must   cause   some,  local 
and  irregular  changes  of  temperature,  changes  in  the 
upper  regions  of  the  atmosphere  as  well  as  near  the 
surface.    Such  abnormal  conditions  of  the  atmosphere 
may  well  give  rise  to  abnormal  refractions,  refractions 
in  azimuth  as  well  as  in  altitude.     Abnormal  changes 
in  the  temperature  of  the  column  of  air  through  which 
the  rays  passed  amounting  to  only  a  very  few  degrees 
would  change  the  course  of  the  rays  by  an  amount 
larger  than  the  sought- for  Einstein  effect.     And  such 
abnormalities  can  neither  be  computed  nor  allowed  for. 
Now  a  photograph  of  the  stars  taken  at  the  time  of 
an  eclipse  cannot,  by  itself,  give  any  result.     It  must 
be  compared  with  a  similar  photograph,  taken  with 
the  same  instrument  at  some  season  of  the  year  when 
the  sun  is  in  a  different  portion  of  the  sky.     Such 
plates  were  taken  at  Sobral  during  the  month  of  July, 
or  some  fifty  days  after  the  eclipse ;  when,  in  the  early 
morning  hours,  the  altitude  of  the  star  group  was 
approximately  the  same  as  on  the  day  of  the  eclipse. 
During  this  interval  of  some  fifty  days  the  telescopes 
were  left  on  their  mountings  and  in  place,  but  the 


208    Gravitation  versus  Relativity 

mirrors  and  the  driving  clocks  were  dismounted  and 
put  into  a  house  to  avoid  exposure.  Thus  the  com- 
parison plates  were  taken  under  almost,  but  not 
identically,  the  same  instrumental  conditions  as  were 
the  eclipse  plates.  Of  course  the  atmospheric  condi- 
tions were  different,  the  temperature  during  the  time 
of  the  eclipse  averaging  some  82°.7F.,  while  it  was  only 
72°.9F.  at  the  time  the  check  plates  were  taken.  But  on 
the  whole,  the  conditions,  instrumental  and  atmos- 
pheric, were  as  near  alike  at  the  times  of  taking  the 
Sobral  eclipse  and  comparison  plates  as  it  was  possible 
to  have  them. 

The  case  is  very  different  for  the  Principe  expedi- 
tion: the  comparison  plates  were  taken  at  Oxford  in 
January  and  February,  under  radically  different  con- 
ditions of  altitude  and  temperature.  And  further,  during 
the  interval  between  the  times  of  taking  these  com- 
parison plates  and  the  day  of  the  eclipse,  the  instru- 
ment had  been  completely  dismantled,  transported 
some  thousands  of  miles,  and  remounted  in  an  entirely 
different  manner.  Any  results,  credited  to  the  Principe 
expedition,  are,  therefore,  subject  to  all  the  uncertain- 
ties due  to  such  radical  differences  of  instrumental 
and  atmospheric  conditions. 

Now  to  discuss  the  eclipse  plates  themselves.  At 
Principe  clouds  seriously  interfered  with  the  work. 
In  the  morning  there  was  a  heavy  thunder-storm, 
and  from  that  time  on  the  sun  was  obscured  by  thick 


The  Eclipse  Plates  209 

clouds,  which  gradually  grew  thinner.  Half  an  hour 
before  totality  the  sun  could  be  glimpsed  occasionally, 
and  by  the  time  the  critical  moments  arrived  it  could 
be  seen  continuously  through  the  drifting  clouds.  The 
time  of  totality  had  been  carefully  computed,  and  when 
this  moment  arrived  the  programme  of  exposing  plates 
was  carried  out.  In  all  sixteen  (16)  plates  were  ob- 
tained, giving  fine  pictures  of  the  inner  corona  and 
of  a  remarkable  prominence  on  the  edge  of  the  sun. 
But  star  images  appeared  on  only  seven  (7)  of  the 
sixteen  plates,  and  these  seven  varied  greatly :  on  some 
of  them  the  images  are  noted  as  "good" ;  on  others  as 
"diffused"  or  "very  faint."  For  the  practical  deter- 
mination of  the  sought-for  "light  deflection,"  the 
images  of  three  stars,  Nos.  3,  4,  and  5,  are  necessary, 
and  the  images  of  these  three  stars  appear  together  on 
only  four  (4)  plates;  on  two  of  which  No.  5  is  noted 
as  "faint,  diffused."  This  left  only  two  plates  out  of 
the  sixteen  as  likely  to  give  trustworthy  results. 
Measures  were  made  on  the  other  plates,  but  the  images 
gave  discordant  results. 

Thus  the  so-called  Principe  results  depend  solely 
upon  two  (2)  plates  taken  under  very  unfavorable 
conditions. 

At  Sobral,  as  has  been  seen,  there  were  two  instru- 
ments, the  regular  1 3-inch  astrographic  lens  and  the 
4-inch  auxiliary  camera.  On  account  of  defects,  which 
had  been  discovered  in  the  large  coelostat  mirror,  the 
14 


210    Gravitation  versus  Relativity 

astrographic  lens  was  stopped  down  to  8-inches  aper- 
ture. On  the  morning  of  the  eclipse  day  the  sun  was 
covered  with  clouds :  at  the  time  of  first  contact  the 
sun  was  invisible.  As  the  eclipse  progressed  towards 
the  total  stage,  however,  the  clouds  diminished,  and 
during  the  critical  period  of  totality  the  region  of  the 
sky  in  the  vicinity  of  the  sun  was  practically  free  from 
clouds.  During  the  middle  of  this  period,  unfortu- 
nately, a  thin  cloud  passed  over  the  sun ;  not  sufficiently 
thick  to  conceal  the  corona,  but  heavy  enough  to  pre- 
vent photography  of  the  stars.  With  the  astrographic 
telescope  nineteen  (19)  plates  were  taken,  with  alter- 
nate exposures  of  5  and  10  seconds;  with  the  4-inch 
camera  eight  (8)  plates  were  taken,  with  a  uniform 
exposure  of  28  seconds.  When  these  plates  were  de- 
veloped, it  was  found  that  at  least  seven  stars  appeared 
on  sixteen  (16)  of  the  plates  taken  with  the  large  tele- 
scope, and  upon  seven  (7)  of  the  plates  taken  with 
the  4-inch  camera.  The  remaining  plates,  taken 
through  the  thin  film  of  cloud,  did  not  show  the 
necessary  star  images. 

The  plates  taken  with  the  astrographic  lens  were  not 
good.  This  was  seen  as  soon  as  the  first  plates  were 
developed,  as  the  following  note  will  show: 

"May  3o,  3  a.  m.,  four  of  the  astrographic  plates  were  de- 
veloped, and  when  dry  examined.  It  was  found  that  there 
had  been  a  serious  change  of  focus,  so  that,  while  the 
stars  were  shown,  the  definition  was  spoilt.  This  change 


The  Eclipse  Plates  211 

of  focus  can  only  be  attributed  to  the  unequal  expansion 
of  the  mirror  through  the  sun's  heat.  The  readings  of  the 
focussing  scale  were  checked  next  day,  but  were  found  un- 
altered at  unim.  It  seems  doubtful  whether  much  can  be 
got  from  these  plates." 

The  results  justified  this  note,  which  was  made  at 
the  time.  The  images  on  all  the  plates  were  diffused 
and  apparently  out  of  focus.  The  exact  cause  of  this 
apparent  change  of  focus  could  not  be  determined, 
but  it  was  attributed  by  the  astronomers  of  the  expe- 
dition to  the  action  of  the  sun's  heat  upon  the  1 6-inch 
mirror,  which  reflected  the  sun  and  its  surroundings 
into  the  lens  of  the  telescope.  The  plates,  however, 
were  all  duly  measured,  but  the  results  were  unsatis- 
factory and  were  given  but  little  weight  by  the  astrono- 
mers in  their  final  conclusions.  The  Report  sums 
up  the  value  of  these  plates  in  the  words : 

"The  photographs  taken  with  the  astrographic  telescope 
support  those  obtained  by  the  f4-inch'  to  the  extent  that 
they  show  considerable  outward  deflection,  but  for  the 
reasons  already  given  are  of  much  less  weight" 

When,  therefore,  the  "Sobral  Results"  are  alluded 
to  one  must  understand  that  the  results  obtained  from 
the  seven  plates  taken  with  the  4-inch  camera  are  alone 
meant.  One  of  these  is  shown  in  the  accompanying 
plate,  which  has  been  reproduced  from  the  plate  con- 
tained in  the  Report.  It  is  from  an  untouched  negative, 


212    Gravitation  versus  Relativity 

but,  of  course,  suffers  in  the  double  reproduction.  The 
disc  of  the  moon  appears  white,  and  the  brilliant 
corona,  which  surrounds  the  concealed  sun,  is  shown 
black,  in  reverse.  The  images  of  the  stars  are  the 
minute  black  specks,  between  the  dashes.  These  dashes 
are  the  only  modifications  of  the  original  plate,  and 
they  were  put  on  solely  to  call  attention  to  the  various 
stars,  which  otherwise  would  escape  notice.  It  will 
be  noted  that  the  image  of  one  star  is  clearly  within 
the  limits  of  the  corona,  itself;  while  two  more  are 
on  the  border,  where  the  plate  appears  faintly  dis- 
colored by  the  light  of  the  corona. 

This  plate  and  the  six  similar  ones  were  measured 
in  the  manner  heretofore  described,  and  similar  meas- 
ures were  made  on  seven  comparison  plates,  taken 
during  July.  After  correcting  the  measures  for  differ- 
ential refraction,  aberration,  orientation,  and  change  of 
scale,  the  mean  or  average  deviation  from  the  standard 
point  of  reference  was  found  for  the  seven  images  of 
each  star  on  the  eclipse  plates;  and  the  similar  mean 
deviation  for  the  seven  corresponding  images  on  the 
seven  comparison  plates.  Assuming  that  the  mean  posi- 
tion, or  deviation,  as  shown  on  the  comparison  plates 
represents  the  true  position,  or  deviation  of  the  star, 
then  the  difference  between  this  mean  and  that  for  the 
eclipse  plates  gives  the  shift  in  position,  or  the  ' 'light 
deflection,"  due  to  the  presence  of  the  sun.  All  of  this  is 
shown  on  the  following  diagram,  where  the  compara- 


Plate  3. 
The  Eclipse  of  the  Sun:  a  photograph  taken  at  Sobral,  Brazil,  May  29,  1919. 

This  is  an  untouched  reproduction  of  the  original  negative  except  for  the  dashes,  which 
were  put  on  to  call  attention  to  the  minute  star  images.  The  Corona,  which  consists  of 
very  tenuous  matter  surrounding  the  Sun,  is  clearly  seen  extending  out  to  and  enveloping 
the  nearest  star.  Einstein  neglects  this  matter  entirely  in  all  his  theories,  and  claims 
that  it  can  have  no  refractive  effect  upon  the  light  which  passes  through  it,  and  no  effect 
upon  the  motion  of  Mercury. 


The  Eclipse  Plates  213 

tive  positions  of  each  one  of  the  stars,  as  it  appears  on 
all  fourteen  plates,  are  shown.  The  small  dots  represent 
the  various  positions  of  the  star  as  deduced  from  each 
one  of  the  seven  eclipse  plates,  the  number  beside  each 
dot  is  the  number  assigned  to  the  plate  in  the  Report. 
The  point  of  the  heavy  arrow  is  the  mean  or  average 
position,  as  determined  from  all  the  plates.  Similarly, 
in  each  figure,  the  small  crosses  represent  the  positions 
of  the  same  star  as  found  on  the  seven  comparison 
plates :  the  notch  of  the  arrow,  the  mean  position.  The 
observed  deflection  is  the  difference  between  the  two 
mean  positions,  and  is  shown  by  a  heavy  arrow :  the 
theoretical  Einstein  shift  is  shown  by  a  broken  arrow. 

In  order  to  bring  out  more  clearly  the  relative  ac- 
curacy of  the  eclipse  and  comparison  plates,  circles  are 
drawn  enclosing  for  each  star  the  eclipse  positions  and 
the  comparison-plate  positions,  respectively.  The  circles 
for  the  eclipse  plates  are  drawn  with  a  full  line ;  those 
for  the  comparison  plates  with  a  broken  line.  It  will 
be  noted  that,  with  a  couple  of  exceptions,  the  com- 
parison plates  are  in  much  better  accord  than  the  eclipse 
plates:  the  groups  of  crosses  being  much  more  con- 
densed than  the  groups  of  dots.  In  three  out  of  the 
seven  cases,  the  circles  overlap. 

The  plotted  results,  Figure  26,  show  clear  evidence 
of  the  effect  of  the  shadow  cone,  heretofore  mentioned, 
upon  the  observed  positions  of  the  stars.  Plate  No.  i 
was  taken  at  the  beginning  of  totality;  Plate  No.  8, 


214    Gravitation  versus  Relativity 

just  before  the  eclipse  ended.  In  the  case  of  every 
star,  the  observed  or  measured  deflection  is  smaller  on 
Plate  i  than  on  Plate  8,  and  in  most  cases  this 
difference  is  extremely  marked.  This  increase  in  the 
measured  deflection  from  Plate  i  to  Plate  8  is  very 
clearly  brought  out  by  dividing  the  plates  into  three 
groups,  and  by  taking  the  average  deflection  of  all  the 
stars  on  all  the  plates  in  each  group.  The  first  group 
contains  Plates  i  and  2;  the  second,  or  intermediate, 
group  Plates  3,  4,  and  5;  the  final  group  the  last  two 
plates,  Nos.  7  and  8.  The  average  deflections,  thus 
found,  are: 


Plates  i  and  2  0^.30 
Plates  3,  4  &  5  0^.34 
Plates  7  and  8  0^.38 


The  general  average  of  all  the  stars  on  all  the  plates 
is  0^.34.  There  thus  appears  to  be  an  average  in- 
crease in  the  size  of  the  observed  deflection  of  some 
2J%  as  the  eclipse  progressed. 

Again,  the  directions  of  the  observed  shifts  show 
changes  with  the  progress  of  the  shadow  cone;  there 
are  marked  differences  between  the  first  and  last 
groups  of  plates  in  the  cases  of  six  out  of  the  seven 
stars. 

For  each  star  there  are  seven  positions,  one  from  each 
of  the  seven  eclipse  plates,  and  seven  comparison  posi- 
tions from  the  seven  comparison  plates.  From  each 


'  No.  3 


\          \      r       ' 
«'    /Va6VlU'' 


No.lOj 


Fig.  26.    Individual  Results  of  Eclipse  and  Comparison  Plates. 

215 


216    Gravitation  versus  Relativity 

possible  combination  of  an  eclipse  position  with  a  com- 
parison position  could  be  derived  an  observed  deflec- 
tion :  in  all  there  are  forty-nine  such  combinations  for 
each  star.  These  forty-nine  combinations,  or  observa- 
tions differ  very  materially  among  themselves.  Some 
of  these  actual  observations  are  shown  in  the  annexed 
diagram :  some  for  star  No.  5,  which  agrees  most  nearly 
with  the  Einstein  prediction,  and  some  for  star  No. 
n,  which  gives  the  most  discordant  results.  In  each 
case  the  thin  lines  show  the  actual  observations,  the 
heavy  broken  arrow  the  predicted  Einstein  effect,  and 
the  full  heavy  arrow,  the  mean  observed  deflection  as 
given  in  the  report  of  the  British  astronomers.  In 
order  not  to  complicate  the  figures  too  greatly,  only 
a  few  of  the  forty-nine  observations  are  shown  in  each 
case.  The  figure  for  star  No.  5  shows  that  the  direc- 
tions of  the  individual  observations  differ  by  approxi- 
mately 36°  on  each  side  of  the  Einstein  arrow,  and  that 
the  observed  amounts  differ  from  o".2O  to  i".O3,  or  by 
a  trifle  over  500%.  The  figure  for  star  No.  n  shows 
that  the  directions  vary  all  the  way  from  -  135°  to  + 
160°,  or  a  total  variation  in  direction  of  practically 
300°;  the  distances  vary  from  o".io  to  o".cp. 

Further,  a  glance  at  these  diagrams  will  show  how 
the  omission  of  one  or  two  plates  would  radically 
change  the  average  or  mean  result,  as  given  in  the 
Report.  In  the  case  of  star  No.  5,  one  of  the  com- 
parison positions  is  extremely  discordant:  if  this  one 


The  Eclipse  Plates 


217 


plate  be  omitted  and  the  average  of  the  remaining  plates 
taken,  then  the  supposed  observed  deflection  would  be 


No.  5  -  The  best  Star 


No.  II  -  The  worst  Star 

Fig.  27.   The  Discordant  Results :  Stars  5  and  1 1. 

greatly  shortened,  and  the  direction  of  the  apparent 
shift  would  be  changed,  so  as  to  differ  radically  from 
the  Einstein  direction.  In  the  case  of  star  No.  n,  if 


218    Gravitation  versus  Relativity 

some  of  the  plates  be  omitted,  the  direction  of  the 
observed  shift  would  be  almost  directly  opposite  to  that 
predicted  by  Einstein. 

The  British  astronomers  selected  seven  out  of  thirty- 
three  plates  and  obtained  from  these  seven  their  results. 
If  this  process  of  selection  be  pushed  still  further  and 
only  two  or  three  plates  taken,  then  almost  any  desired 
result  could  be  obtained :  the  deflection  could  be  made  to 
appear  almost  anything,  in  direction  and  in  amount. 

The  mean  results  from  all  the  plates  are  summarized 
in  the  following  table,  which  is  taken  directly  from  the 
Report.  In  this  table  the  stars  are  arranged  in  the 
order  of  their  respective  distances  from  the  sun,  and 
the  radial  component  of  the  observed  deflection  is  given 
in  each  case. 

TABLE  III 

Radial  Displacement  of  Individual  Stars 
Star      Calculation         Observation 

II  0".32  0".20 

10  o".33  o".32 

6  o".40  o"-56 

5  o".53  o".54 

4  o".75  o".84 

2  o".8s  o".97 

3  o".88  i".02 

This  table  shows  that,  on  the  average,  the  observed 
deflection,  as  given  by  the  British  astronomers,  differs 
by  19%  from  the  calculated  Einstein  value.  In  the 


The  Eclipse  Plates  219 

cases  of  two  stars  the  agreement  between  theory  and 
observation  is  nearly  perfect,  the  observed  value  being 
only  3%  in  error:  in  other  cases,  however,  the  dif- 
ferences range  from  11%  to  60%.  Not  only  are  the 
actual  amounts  of  the  deflections,  as  observed,  for  the 
individual  stars,  thus  different  from  the  theoretical 
Einstein  values,  but  the  rate  of  decrease  from  star  to 
star  is  radically  different  from  that  predicted.  The 
difference  between  the  deflection  of  the  star  nearest  the 
sun  and  that  of  the  farthest  star  should  be,  according  to 
Einstein,  0^.56;  while  the  observed  or  measured  dif- 
ference was  o".82,  practically  50%  out  of  the  way. 

The  diagrams,  given  above,  show  clearly  that  the 
observed  displacements  of  the  stars  do  not  agree  in 
direction  with  the  predicted  Einstein  effect.  This  point 
was  nowheres  mentioned  in  the  Report,  which  took  up 
only  the  amount  of  the  radial  component  of  the  actual 
displacement.  But,  after  the  measurements  of  the 
plates  became  available  for  study,  several  investigators 
called  attention  to  this  fact  of  a  radical  disagreement 
in  direction  between  the  observed  and  predicted  dis- 
placements. These  differences  in  direction  amount  to 
many  degrees,  in  the  case  of  the  star  farthest  from  the 
sun  to  37°.  Thus,  even  the  seven  best  plates  out  of 
the  thirty-three,  which  showed  star  images,  give  incon- 
sistent results: — the  observed  shifts  in  the  star  images, 
if  real,  do  not  coincide  with  the  Einstein  effect  either 
in  amount  or  in  direction. 


220    Gravitation  versus  Relativity 

But  these  seven  plates  clearly  indicate  shifts,  or 
displacements,  of  the  star  images,  and,  in  a  very  general 
way,  these  shifts  agree  somewhat  both  in  direction  and 
amount  with  the  predicted  Einstein  effect.  It  has  been 
claimed  by  many  that  the  differences  between  the 
observed  and  predicted  shifts  are  no  greater  than  should 
be  expected;  that  the  differences,  as  shown  in  the 
diagrams,  are  the  mere  unavoidable  errors  of  measure- 
ment and  calculation  which  creep  into  every  physical  or 
astronomical  research.  If  these  differences  are  within 
the  limits  of  allowable  accidental  errors,  and  if  the  dis- 
placements are  real,  then  these  plates  certainly  furnish 
some  prima  facie  evidence  in  favor  of  the  relativity 
theory.  But,  if  these  differences  are  real  and  greater 
than  allowable  accidental  errors,  then  the  case  is  altered, 
and  the  evidence  is  nowheres  near  as  strong,  for  then 
there  is  a  distinct,  proved  disagreement  between 
observation  and  prediction.  Now  this  very  question 
was  investigated  by  Dr.  Henry  Norris  Russell,  of 
Princeton  University,  a  most  ardent  upholder  of  the 
relativity  theory.  He  studied  these  star  displacements 
with  a  view  of  determining  whether  the  departures 
from  the  Einstein  predicted  effects  are  real  or  not,  and, 
if  real,  of  finding  some  possible  explanation  of  them. 
As  a  result  of  an  exhaustive  examination  of  them, 
he  concludes  that  these  differences  between  the  observed 
and  the  predicted  displacements,  these  non-Einstein  dis- 
placements, as  he  calls  them,  are  real,  and  cannot  be 


The  Eclipse  Plates  221 

attributed  to  mere  accidental  errors  of  observation  and 
measurement.  He  finds  that  the  Einstein  effect  alone 
will  not  represent  the  observations;  that,  in  order  to 
represent  satisfactorily  the  measurements  from  the 
plates,  it  is  necessary  to  supplement  the  Einstein  predic- 
tion by  a  distortion  of  some  kind. 

Dr.  Russell  assumes  that  the  most  probable  source 
of  these  proved  non-Einstein  deflections  is  to  be  found 
in  instrumental  errors:  in  an  alteration  in  the  shape 
of  the  mirror,  caused  by  the  heat  of  the  sun.  He 
finds  from  an  elaborate  investigation  the  form  and 
amount  of  mirror  distortion  that  will  best  satisfy  these 
non-Einstein  effects,  and  concludes  that,  if  the  mirror 
had  been  distorted  into  a  cylindrical  form  with  a  definite 
radius  of  curvature,  then  the  non-radial  displacements 
can  be  well  accounted  for.  The  sun's  heat  undoubtedly 
distorted  the  mirror,  and  it  is  highly  probable  that 
some  portion  of  the  observed  displacement,  radial  as 
well  as  non-radial,  was  due  to  such  distortion.  The 
mirror  of  the  large  instrument  was  so  affected,  and 
the  nineteen  plates  taken  with  that  instrument  were 
very  unsatisfactory,  and  were  practically  discarded  by 
the  British  astronomers.  It  has  been  known  for  many 
years  that  the  horizonal  form  of  instrument  is  utterly 
unsuitable  for  exact  measurements;  a  fact  tacitly 
admitted  in  the  concluding  paragraphs  of  the  Report. 

But  one  point  is  perfectly  clear.  If  it  be  admitted 
that  the  heat  of  the  sun  so  distorted  the  mirror  of  the 


222    Gravitation  versus  Relativity 

apparatus  as  to  cause  errors  of  20%,  in  some  cases  of 
50%,  of  the  measured  displacement,  then  the  entire  set 
of  plates  is  worthless  for  proving  the  existence  or  non- 
existence  of  the  "Einstein  effect."  Following  the 
methods  used  by  Russell  it  might  not  be  impossible  to 
find,  by  suitable  adjustments  of  radii  of  curvature  and 
axes,  a  distorted  shape  of  the  mirror  that  would  explain 
the  entire  observed  deflections. 

Either  these  observations  represent  actual  displace- 
ments of  the  star  images,  or  they  represent  mere 
instrumental  distortions:  either  they  are  good  and 
must  be  explained  in  their  entirety ;  or  they  are  worth- 
less and  must  be  discarded.  If  these  deflections  of  the 
star  images  on  the  plates  are  real,  and  are  not  due  to 
distortions  of  the  photographic  film,  or  to  defects  in 
the  photographic  apparatus,  then  Russell  has  shown 
clearly  that  they  cannot  be  explained  in  their  entirety 
by  the  Einstein  theory.  That  theory  will  account  for 
deflections  in  the  star  images,  but  not  for  the  observed 
deflections. 

The  Einstein  theory  calls  for  a  deflection  of  light 
amounting  to  i"-75  at  the  edge  of  the  sun,  and  decreas- 
ing proportionally  with  the  distance  from  the  centre, 
so  that  for  a  ray  passing  the  sun  at  a  distance  of  one 
radius  from  the  edge,  or  two  radii  from  the  centre, 
the  deflection  should  be  exactly  one-half  the  above 
amount.  From  this  law  of  decrease  and  the  known 
distances  of  the  star  images  from  the  centre  of  the 


The  Eclipse  Plates  223 

sun,  the  observed  deflections  can  readily  be  transformed 
into  what  they  would  have  been  at  the  sun's  edge. 
This  is  done  in  the  Report,  and  the  average  result  of 
all  the  stars  taken.  In  finding  this  mean,  the  measure- 
ments in  declination  were  given  twice  the  weight  of 
those  in  right  ascension.  The  Report  gives  these  final, 
or  mean,  results  for  the  three  separate  sets  of  observa- 
tions as- 

From  the  seven  (7)  plates  taken  at  Sobral  with  the 
4-inch  camera, 

I  ".98 
with  a  probable  error  of  about  ±  o".i2. 

From  the  two  (2)  plates  taken  at  Principe  with  the 
13-inch  astrographic  lens 

i".6i 
with  a  probable  error  of  about  ±  o".3o. 

From  the  sixteen  ( 16)  plates  taken  at  Sobral  with  the 
13-inch  astrographic  lens 

o"-93 

"For  reasons  already  described  at  length  not  much  weight 
is  attached  to  this  determination."  * 

These  results,  in  each  case,  are  the  means  of  the 
radial  components  only;  nothing  whatever  being  given 
as  to  the  directions  in  which  the  actual  displacements 
took  place.  The  Einstein  theory  requires  a  deflection, 
not  only  of  a  certain  definite  amount,  but  also  in  a 

*  Report. 


224    Gravitation  versus  Relativity 

certain  definite  direction.  To  discuss  the  amount  of 
the  observed  deflection  is  to  discuss  only  one-half  of 
the  whole  question,  and  the  less  important  half  at  that. 
The  observed  deflection  might  agree  exactly  with  the 
predicted  amount,  but,  if  it  were  in  the  wrong  direc- 
tion, it  would  disprove,  not  prove,  the  relativity  theory. 
You  cannot  reach  Washington  from  New  York  by 
travelling  north,  even  if  you  do  go  the  requisite  num- 
ber of  miles.  Now,  the  diagrams,  above  given,  of 
the  seven  best  plates,  the  seven  taken  at  Sobral  with 
with  4-inch  camera,  show  clearly  and  definitely  that 
the  observed  deflections  are  not  in  the  directions  re- 
quired by  the  Einstein  theory.  In  the  case  of  star 
No.  10,  the  observed  mean  deflection  differs  in  direc- 
tion by  some  28°  from  the  predicted  direction:  not 
only  that,  but  every  one  of  the  seven  plates  shows  the 
star  deflected  in  the  same  direction  from  that  called 
for  by  the  relativity  theory;  every  star  dot,  in  the 
diagram,  lies  on  the  same  side  of  the  Einstein  arrow. 
Similarly  for  star  No.  n,  every  dot  again  lies  on  the 
same  side  of  the  Einstein  arrow,  and  the  mean  deflec- 
tion differs  by  37°  from  the  predicted.  In  this  case 
two  of  the  individual  plates  give  deflections  practically 
in  the  reverse  direction  to  that  called  for  by  the  theory. 
The  best  agreement  between  theory  and  observation 
is  given  by  star  No.  4,  where  the  mean  difference 
amounts  to  about  a  single  degree:  but,  even  in  this 
case,  the  individual  results  differ  by  as  much  as  30°. 


The  Eclipse  Plates  225 

The  relativitist  either  totally  disregards  these  dis- 
cordances in  the  directions  of  the  observed  deflections, 
or  invokes  the  heating  effect  of  the  sun  to  distort  the 
mirror  by  just  the  proper  amount  to  explain  them 
away! 

Again,  disregarding  directions  entirely,  and  taking 
into  account  only  the  size  of  the  deflection,  it  is  noted 
that  the  disagreement  between  the  three  mean  results, 
as  given  in  the  Report,  is  over  100%  ;  the  largest  value 
being  well  over  twice  that  of  the  smallest.  The  actual 
amount  of  the  deflection  as  obtained  with  the  astro- 
graphic  lens  is  58%  of  that  obtained  at  Principe  and 
only  47%  of  that  of  the  4-inch  camera  at  Sobral. 
This  difference  in  results  is  far  beyond  the  limits  of 
accidental  errors.  Distortions  in  the  mirror  of  the 
large  coelostat,  due  to  the  heating  effects  of  the  sun, 
are  called  upon  to  explain  this  discordance.  But,  in 
this  case  such  procedure  is  absolutely  justifiable,  for 
the  effects  of  the  distortions  were  noted  in  the  nega- 
tives, before  any  of  the  plates  were  measured.  But  the 
fact  that  one  set  of  plates  was  ruined  by  such  heat- 
ing effects,  renders  the  other  set,  taken  under  similar 
conditions,  at  least  of  doubtful  value. 

When  the  deflections  of  light,  as  actually  observed, 
are  considered  both  in  direction  and  in  amount,  the 
discordances  with  the  predicted  Einstein  effect  become 
marked,  and  the  plates  present  little  or  no  evidence  to 
support  the  relativity  theory.  Further,  if  these  defleo 
15 


226    Gravitation  versus  Relativity 

tions  are  real,  and  not  due  to  instrumental  errors  (so 
readily  called  upon  by  the  relativitist  to  explain  every- 
thing that  the  relativity  theory  cannot  account  for), 
then  it  has  not  yet  been  shown  that  the  relativity  theory 
is  the  only  possible  explanation.  As  a  matter  of  fact 
there  are  other  perfectly  possible  explanations  of  a 
deflection  of  a  ray  of  light;  explanations  based  upon 
e very-day,  common-place  grounds.  Abnormal  refrac- 
tion in  the  earth's  atmosphere  is  one;  refraction  in 
the  solar  envelope  is  another.  The  atmospheric  con- 
ditions under  which  the  eclipse  plates  were  taken  were 
necessarily  abnormal,  and  the  plates,  themselves,  clearly 
show  that  the  rays  of  light  passed  through  a  mass  of 
matter  in  the  vicinity  of  the  sun;  a  mass  of  density 
sufficient  to  clearly  imprint  its  picture  upon  the  photo- 
graphic plates. 

Such  is  the  evidence,  such  are  the  observations, 
which,  according  to  Einstein,  "confirm  the  theory  in 
a  thoroughly  satisfactory  manner/' 


CHAPTER  VIII 

THE  OBSERVED  PHENOMENA  AND  CLASSICAL  METHODS 

THE  former  chapters  show  clearly  that  the  Rela- 
tivity Theory  is  inadequate  to  explain  either  the 
observed  motions  of  the  planets,  or  the  observed 
deflections  of  light  rays :  it  can  account  for  the  motion 
of  the  perihelion  of  Mercury  and  for  a  certain  definite 
deflection  of  light,  but  it  cannot  account  for  other 
observed  motions  of  Mercury  and  Venus,  nor  for  the 
light  deflections  as  actually  observed.  The  relativitist 
is  forced,  either  to  deny  the  existence  of  these  other 
motions,  or  to  supplement  his  theory  by  some  other 
agency  to  account  for  those  things,  which  the  relativity 
theory  by  itself  cannot  explain.  In  the  words  of  the 
mathematician,  the  relativity  theory,  alone,  is  not 
sufficient. 

On  the  other  hand,  the  ordinary  classical  methods 
of  physical  and  astronomical  research  can  fully  explain 
all  the  observed  phenomena ;  the  motions  of  the  planets 
can  be  fully  accounted  for  by  the  presence  of  matter 
known  to  exist,  and  the  light  deflections,  if  real,  can 

227 


228    Gravitation  versus  Relativity 

be  explained  as  the  result  of  refraction  through  the 
cosmic  dust  surrounding  the  sun.  It  is  true  that  New- 
comb,  in  forming  his  tables  of  planetary  motion  for 
the  Nautical  Almanac,  dismissed  the  idea  of  unknown 
masses  of  matter  and  introduced  arbitrary  corrections 
to  take  care  of  the  observed  discordances;  corrections, 
however,  so  introduced  as  to  commit  him  to  no 
hypothesis  as  to  the  cause  of  the  observed  excess 
motions.  Newcomb  was  largely  influenced  by  the  desire 
for  simplicity,  or  rather  a  desire  to  avoid  a  "com- 
plication in  the  tabular  theory."  His  investigations, 
heretofore  fully  outlined,  showed  that  no  one  single 
planet,  nor  a  single  group  of  planetoids,  could  fully 
account  for  all  the  observed  discrepancies,  although 
a  group  of  planetoids  between  Mercury  and  Venus 
more  nearly  satisfied  the  conditions  than  any  other 
hypothesis. 

THE  MOTIONS  OF  THE  PLANETS: 

Simplicity  may  be  desirable,  but  it  is  not  essential. 
It  was  undoubtedly  the  desire  for  simplicity,  the  wish 
to  find  a  single  cause  for  the  sometimes  conflicting 
motions,  that  led  Newcomb  and  other  investigators 
into  their  seeming  difficulties.  Yet  it  is  hardly  likely 
that  all  the  six  or  seven  small  discordant  motions  of 
the  various  planets  have  the  same  cause.  There  are 
several  possible,  even  probable,  causes  for  each  and 
every  one  of  the  discordances,  and  the  true  explanation 


Classical  Methods  229 

probably  lies,  not  in  a  single  cause,  but  in  a  combina- 
tion of  causes ;  not  in  a  single  unknown,  unseen  planet, 
or  a  single  group  of  planets,  but  in  a  combination  of 
groups,  or  an  irregularly  scattered  mass  of  matter 
about  the  sun.  The  sun  is  known  to  be  non-homo- 
geneous, and  it  is  known  to  be  surrounded  by  a  vast 
irregular  mass  of  matter,  by  an  envelope  extending 
far  beyond  the  orbit  of  the  earth.  In  the  fact  of  this 
irregularity  of  the  sun  itself  and  in  the  presence  of 
this  matter  can  be  found  full  and  satisfactory  explana- 
tions of  all  the  various  motions. 

The  sun  is  known  to  be  a  rotating,  cooling  mass 
of  gas,  and  fundamental  laws  of  physics  show  that 
such  a  body  must  be  a  spheroid.  The  equatorial  dia- 
meter of  the  sun  must,  thus,  exceed  the  polar  by  an 
amount  not  less  than  o".O5,  a  quantity  far  below  the 
limits  of  possible  measurement.  As  has  been  seen  in 
a  former  chapter,  the  actual  measurements  indicate 
an  ellipticity  somewhat  larger  than  this,  an  ellipticity 
just  verging  on  the  limits  of  measurement  at  around 
o".io.  Such  an  ellipticity  in  the  sun  would  cause  a 
motion  of  7/'o  in  the  perihelion  of  Mercury,  and  cor- 
responding, although  very  much  smaller,  motions  in 
the  perihelia  of  the  other  planets.  This  is  certainly  a 
possible,  rather  an  extremely  probable,  cause  of  a  con- 
siderable portion  of  the  unexplained  discordance.  And 
there  is  nothing  at  all  new  or  unique  about  such  an 
explanation.  It  has  been  in  use  for  well  over  one 


230    Gravitation  versus  Relativity 

hundred  and  fifty  (150)  years  to  explain  and  account 
for  similar  motions  among  the  satellites  of  Jupiter. 
The  disc  of  Jupiter  is  clearly  elliptical,  and  as  early  as 
1748  Euler  showed  that  such  elliptical  figure  would 
cause  irregularities  in  the  motions  of  the  satellites,  and 
in  1758  Walmsley  showed  that  it  would  produce  a 
rotation  of  the  orbit  of  the  satellite  precisely  similar 
to  the  now  much  discussed  motion  of  Mercury.  Since 
that  date  it  has  been  an  accepted  method  of  Celestial 
Mechanics  to  account  for  perihelial  motions  of  the 
various  satellites  of  the  Solar  System  by  means  of  the 
observed  and  measured  ellipticity  of  the  planet  about 
which  they  revolve.  The  elliptic  shape  of  the  earth 
accounts  for  several  of  the  observed  irregularities  in 
the  motion  of  the  moon. 

But  the  elliptic  shape  of  the  sun  will  not  account 
for  more  than  a  small  portion  of  the  discordance  in 
the  motion  of  Mercury's  perihelion,  and  will  not  recon- 
cile the  observed  differences  in  the  other  motions  of 
both  Mercury  and  Venus.  Such  remaining  discord- 
ances must  have  other  causes,  and  such  causes  can  be 
found  in  the  immense  envelope  of  matter,  which  sur- 
rounds the  sun  and  which  extends  far  beyond  the  orbit 
of  the  earth.  Unfortunately  the  exact  distribution  of 
this  matter  throughout  space  is  unknown;  and,  there- 
fore, its  effects  upon  the  motions  of  the  planets  can- 
not be  accurately  calculated.  While,  thus,  the  problem 
fails  of  a  direct  solution,  yet  it  is  possible  to  attack  it 


Classical  Methods  231 

in  reverse.  That  is,  it  is  possible  to  find  what  distribu- 
tion of  this  matter  would,  under  the  Newtonian  law, 
give  rise  to  the  discordant  motions  as  actually  observed. 
Then  this  calculated  distribution  can  be  compared  with 
the  known  facts  to  determine  whether  or  not  it  con- 
forms to  observation  and  is  within  the  bounds  of 
reason.  This  problem  is  similar  to  that  solved  by 
Adams  and  Leverrier,  which  resulted  in  the  discovery 
of  Neptune.  It  will  be  remembered  that  these  eminent 
astronomers  calculated,  from  certain  observed  irregu- 
larities in  the  motions  of  the  planet  Uranus,  the  posi- 
tions and  motions  of  an  hypothetical  planet,  which 
would  cause  such  motions;  and  that,  subsequently,  a 
planet,  now  known  as  Neptune,  was  found  very  close 
to  the  predicted  place. 

Now  in  a  very  general  way  it  is  known  that  this 
solar  envelope  is  lens-shaped  and  of  very  minute 
density;  more  dense  in  the  immediate  vicinity  of  the 
sun  than  in  its  more  remote  extensions  beyond  the 
orbit  of  the  earth.  Observations  of  the  zodiacal  light 
show  that,  in  the  outer  portions,  the  principal  plane 
of  this  lens  of  matter  does  not  differ  radically  from 
that  of  the  earth's  orbit;  while  photographs  of  the 
corona  show  that,  in  the  immediate  vicinity  of  the 
sun,  this  plane  does  not  differ  greatly  from  that  of 
the  sun's  equator.  This  general  distribution  can  be 
approximated  to  by  assuming  the  whole  mass  to  be 
made  up  of  ellipsoids  of  revolution,  each  ellipsoid  to 


232    Gravitation  versus  Relativity 

be  of  uniform  density,  but  the  larger  ones  to  be  of 
much  less  density  than  the  inner  ones.  Some  such 
assumption  is  necessary  to  reduce  the  problem  to  the 
realm  of  figures:  the  selection  of  ellipsoids  of  revolu- 
tion is  naturally  indicated  by  the  fact  that  all  the 
known  bodies  of  the  solar  system  are  such  ellipsoids. 

An  ellipsoid,  or  ring,  of  matter  wholly  within 
the  orbit  of  a  planet  will  give  a  direct  motion  to  the 
perihelion.  But  if  the  orbit  actually  lies  in  the  matter 
composing  such  ellipsoid,  then  the  effect  is  the  opposite 
and  the  motion  of  the  perihelion  will  be  retrograde. 
This,  of  course,  upon  the  assumption  that  the  density 
is  uniform  throughout;  if  the  density  is  much  greater 
in  the  central  portions  of  the  ellipsoid,  then  the  retro- 
grade effect  of  the  outer  portion  may  be  overcome  and 
the  total  effect  upon  the  perihelion  may  be  direct,  but 
the  motion  will  be  less  than  that  due  to  the  central 
portion  alone.  By  adjusting  the  rate  at  which  the 
density  is  assumed  to  decrease,  any  motion  of  the  peri- 
helion, direct  or  retrograde,  within  limits  can  be 
obtained.  To  changes  in  the  density  of  the  envelope 
surrounding  the  sun  may  thus  be  attributed  the  dis- 
cordant motions  of  the  perihelia  of  the  four  inner 
planets,  and  especially  the  retrograde  discrepancy  in 
the  motion  of  Venus. 

For  purpose  of  computation  the  entire  mass  may 
be  supposed  to  be  made  up  of  three  superimposed 
ellipsoids,  each  of  constant  density.  This  merely  makes 


Plate  4. 

The  Sun,  photographed  in  the  light  of  glowing  Hydrogen  at  Mount  Wilson  Observatory: 
vortex  phenomena  near  the  spots  are  especially  prominent. 

The  so-called  astronomical  proof  of  the  Einstein  theory  is  based  upon  the  assump  Jo  i  that  the  Sun 
is  a  perfect  sphere,  each  concentric  layer  of  which  is  of  uniform  density.  This  photograph  clearly 
shows  that  the  Sun  is  a  mass  of  whirling,  rising  and  falling  gases;  never  at  rest  and  r.ever  uniformly 
distributed.  The  actual  condition  of  the  Sun,  as  shown  by  this  photograph,  is  totally  disregarded  by 
Einstein. 


Classical  Methods  233 

the  changes  in   density  abrupt,   instead  of   gradual. 
These  three  are: — 

a.  A  small  central  ellipsoid  entirely  within  the  orbit  of 
Mercury.    The  position  of  this  ellipsoid  in  space  was 
determined  from  the  discordances  themselves. 

b.  An  intermediate  ellipsoid  entirely  within  the  orbit  of 
the  earth,  but  extending  beyond  the  orbit  of  Venus. 
The  principal  plane  of  this  was  assumed  as  being  the 
same  as  that  of  the  orbit  of  Jupiter. 

c.  An  outer  ellipsoid  entirely  within  the  orbit  of  Mars, 
but  extending  beyond  the  orbit  of  the  earth.     The 
principal  plane  of  this  was  also  assumed  as  being  the 
same  as  that  of  Jupiter's  orbit 

The  effect  of  each  ellipsoid  upon  the  perihelia,  the 
nodes,  and  the  inclinations  of  the  planets  can  be  found 
by  formulas  of  Celestial  Mechanics,  and  the  positions 
and  densities  of  those  ellipsoids,  which  will  best  account 
for  all  the  motions,  can  be  determined.  No  distribu- 
tion can  be  found  that  will  rigorously  satisfy  all  the 
motions,  but  the  positions  and  densities  of  those 
ellipsoids  can  be  found,  which  will  approximately 
satisfy  all  the  equations  and  practically  account  for  all 
the  discordances  in  the  motions  of  the  planets.  The 
annexed  table  shows  with  what  a  high  degree  of  ac- 
curacy the  motions  of  the  planets  can  be  accounted  for 
under  the  action  of  the  sun  and  this  widely  distributed 
matter.  For  purposes  of  comparison  the  residuals  on 
the  basis  of  the  Einstein  theory  are  also  given : 


234    Gravitation  versus  Relativity 

TABLE  IV 
Final  Discordances  in  the  Motions  of  the  Planets 

Amounts  Final  Discordances: 

PERIHELIA:  to  account  for:  Einstein  Poor 

(1)  Mercury  +  39".8  -    3".o  +o".i 
Venus                        —    7".3                 —  is".9             +  o".2 
Earth                        +    5".9                 +    2".i             +  o".3 

(3)  Mars  +    8".i  +    6".8  +  3".i 

NODES: 

Mercury  -f-    5".!  +    5".!  +  o".6 

(2)  Venus  +  io".2  +  io".2  —  i".s 

ECCENTRICITY: 

(4)  Mercury  —    o".9  —  o".9  —  o".2 

The  relative  probabilities  of  two  theories,  or  two 
solutions  of  a  problem,  are  usually  determined  from  the 
final  differences,  or  residuals,  as  these  differences  are 
called.  That  solution  is  deemed  the  more  probable 
which  makes  the  sum  of  the  squares  of  the  residuals 
the  smaller.  If  this  test  be  applied  to  the  residuals  in 
the  above  table,  the  results  are : 

Einstein  theory  473 

Sun  and  Solar  envelope         14 

And  these  clearly  indicate  how  very  much  more  proba- 
ble is  the  explanation  of  the  motions  of  the  planets 
as  due  to  the  presence  of  matter  in  space,  than  as  due 
to  the  hypotheses  of  Einstein. 


Qassical  Methods  235 

Now  in  order  to  determine  whether  or  not  such  a 
distribution  of  matter,  as  called  for  in  the  above  solu- 
tion, is  possible,  some  computations  must  be  made  as 
to  its  necessary  mass  and  density.  The  formulas, 
however,  are  such  that  the  mass  or  density  of  each 
ellipsoid  depends  upon  its  radius,  and  cannot  be  in- 
dependently determined.  The  smaller  the  ellipsoid,  the 
greater  mass  it  must  contain  in  order  to  account  for 
the  various  motions.  But  the  radii  can  be  arbitrarily 
assumed,  and  the  corresponding  masses  and  densities 
found.  Taking  the  shape  of  the  ellipsoid  to  be  such 
that  the  axis  about  which  it  revolves  is  only  i/ioth  of 
the  larger  radius,  and  supposing  the  inner  ring,  or 
ellipsoid,  to  extend  some  forty  (40)  radii  of  the  sun, 
or  approximately  to  one-half  the  distance  of  Mercury, 
and  the  second  ellipsoid  to  be  just  a  trifle  smaller  than 
the  orbit  of  the  earth,  we  have  the  following : 

Mass  Density 

Inner  ellipsoid  3          8.9  X  io~8 

Second  ellipsoid     4/7         1.3  X  io~10 

where  the  density  of  air  is  unity  and  the  masses  are 
given  in  terms  of  that  of  Mercury. 

The  density  of  the  air  is  measured  by  its  pressure, 
and  the  standard  at  sea-level  is  760  mm.  of  mercury. 
The  pressure  of  this  matter  in  space  would  be  measured 
by  one  fifteen  thousandth  (i/i5,oooth)  of  a  millimeter. 


236    Gravitation  versus  Relativity 

Now  to  obtain  some  idea  as  to  what  this  density  really 
means,  suppose  that  one-half  of  the  total  mass  of  the 
inner  ellipsoid  be  concentrated  into  a  ring  of  planetesi- 
mals  of  an  average  diameter  of  fifty  (50)  miles;  the 
outer  edge  of  the  ring  being  at  the  outer  limits  of  the 
ellipsoid,  or  one-half  the  distance  of  Mercury  from  the 
sun.  If  the  cross-section  of  such  a  ring  contained  one 
hundred  (100)  such  planetesimals,  that  is,  if  it  con- 
sisted of  ten  rows  of  bodies,  one  above  the  other,  each 
row  containing  ten,  then  the  average  distance  between 
these  bodies  would  be  some  17,000  miles.  Individually 
such  bodies  would  be  invisible,  for  they  would  never 
be  more  than  15°  from  the  sun,  and  a  body  of  such 
small  diameter  would  be  completely  lost  in  the  glare. 
At  the  same  angular  distance  from  the  sun,  Mercury, 
3000  miles  in  diameter,  is  a  difficult  object.  The  only 
possible  chance  of  discovering  any  one  of  such  a  group 
of  bodies  would  be  at  the  times  of  total  solar  eclipses, 
and  even  at  such  times  the  chances  of  actually  seeing 
such  a  small  body  would  be  almost  infinitesimal. 

In  order  to  account  for  the  motions  of  the  nodes,  the 
inclination  of  such  a  ring,  or  ellipsoid,  of  planetesimals 
would  have  to  be  comparatively  large.  Computation 
shows  that  the  inclination  to  the  ecliptic  of  such  a  ring 
would  be  between  7°  and  8°.  Newcomb,  in  his  discus- 
sion of  the  general  subject  many  years  ago,  considered 
such  a  great  inclination  as  highly  improbable,  believing 
that  such  a  group  of  bodies  would  tend  to  gather 


Classical  Methods  237 

around  a  plane  somewhere  between  that  of  the  orbit  of 
Mercury  and  that  of  the  invariable  plane  of  the  plane- 
tary system;  that  is,  between  7°  and  i°.  The  inclina- 
tion of  Mercury's  orbit  is  7°  and  that  of  sun's  equator 
is  7°  15',  and  it  would  appear  more  reasonable  to 
expect  a  group  of  small  planets  in  the  immediate  vicin- 
ity of  these  two  bodies  to  have  inclinations  somewheres 
near  the  two.  Further,  if  Newcomb's  reasoning  be  cor- 
rect, then  the  great  ring  of  planetoids,  between  Mars 
and  Jupiter,  should  certainly  show  a  distinct  grouping 
near  the  plane  of  Jupitor's  orbit,  for  these  bodies  are  at 
a  remote  distance  from  the  sun  and  Mercury,  and  are  in 
the  immediate  vicinity  of  Jupiter  itself.  Yet  the  indi- 
vidual planetoids,  which  form  this  ring,  have  in- 
clinations varying  all  the  way  up  to  35°.  The  four 
largest  of  the  group  have  inclinations  of  10°,  34°,  7°, 
and  13°  respectively.  Thus,  in  this  matter,  Newcomb's 
reasoning  fails  in  a  case  very  much  more  favorable  to 
his  theory. 

There  is  apparently  no  mechanical  nor  physical 
reason  for  the  non-existence  of  a  group,  or  groups,  of 
bodies,  sufficient  to  explain  all  the  irregularities  in  the 
motions  of  the  planets.  Thus,  all  the  discordances, 
including  that  of  the  perihelion  of  Mercury,  can 
readily  be  accounted  for  by  the  action,  under  the  New- 
tonian law,  of  matter  known  to  be  in  the  immediate 
vicinity  of  the  sun  and  the  planets. 

It  is,  however,  possible  that  the  Einstein  hypotheses 


238    Gravitation  versus  Relativity 

be  true,  and  that  the  discordant  motions  of  the  planets 
result  from  a  combination  of  the  Einstein  motions 
and  the  effect  of  the  widely  distributed  matter  in  space. 
Just  as  a  definite  distribution  can  be  found  which  will 
explain  the  discordances  given  by  Newcomb,  so  also 
another  and  different  distribution  can  be  found  that 
will  more  or  less  fully  account  for  the  discordances 
remaining  after  applying  the  Einstein  effects.  And  this 
distribution  is  not  radically  different  from  that  found 
above,  except  that  the  density  of  the  matter  is  more 
nearly  uniform  throughout  space;  it  is  less  dense  near 
the  sun  and  more  dense  in  the  vicinity  of  the  earth's 
orbit.  In  fact  the  corresponding  ellipsoids  would 
be:— 

Mass         Density 

Inner  ellipsoid        1/4      7.4  X  io~9 
Second  ellipsoid      5/7       1.5  X  io~ I0 

in  which  as  before  the  unit  of  mass  is  Mercury  and  that 
of  density,  air  at  standard  pressure. 

Thus  the  motions  of  the  planets  do  not  prove  the 
truth  of  the  Einstein  theory,  nor,  on  the  other  hand,  do 
they  prove  its  falsity.  While  these  motions  can  be 
accounted  for  by  a  certain  distribution  of  matter  in 
the  solar  envelope,  it  has  not  yet  been  established  by 
observation  that  the  matter  is  distributed  through  space 
in  the  required  way.  In  the  present  state  of  our  knowl- 


Classical  Methods  239 

edge  regarding  this  matter,  the  motions  of  the  planets 
do  not  and  cannot  furnish  a  definite  answer  to  the 
question  as  to  the  validity  of  the  relativity  hypothesis. 
It  is  then  a  problem  of  observational  astronomy  to 
investigate  the  actual  distribution  and  density  of  the 
matter  in  the  solar  lens,  and  to  determine  whether  or 
not  it  approximates  the  conditions  necessary  to  account 
for  the  planetary  motions. 

But  one  conclusion  is  certain,  the  Einstein  hypotheses 
and  formulas  are  neither  necessary  nor  sufficient  to  ex- 
plain the  discordances  in  the  planetary  motions. 

THE  SOBRAL  ECLIPSE  PLATES: 

The  curvature  of  light  rays,  supposed  to  have  been 
clearly  proved  by  the  eclipse  photographs,  may  or  may 
not  exist.  The  seven  Sobral  plates  show  clear  evi- 
dence of  shifts  in  the  star  images,  but  it  has  not  been 
shown  that  such  shifts  are  in  fact  due  to  the  bending 
of  the  rays  at  the  sun.  Such  apparent  shifts  may  be 
due  to  instrumental  errors,  to  distortions  of  the  mir- 
ror, to  abnormal  refraction  in  the  earth's  atmosphere 
caused  by  the  cooling  effect  of  the  passing  shadow 
cone. 

But,  if  the  deflections  are  real  and  are  caused  by  a 
bending  of  the  rays  of  light  at  or  near  the  sun,  such 
bending  may  be  due  to  perfectly  natural  causes. 
The  Sobral  photographs  show  clearly  that  the  rays 
of  light,  in  their  course  from  the  distant  stars,  passed 


240    Gravitation  versus  Relativity 

through  masses  of  matter  near  the  sun.  This  matter 
was  sufficiently  dense  and  reflected  enough  sunlight 
to  imprint  its  image  upon  the  photographic  plates,  and 
there  can  be  no  question  as  to  its  existence  and  its 
presence  in  the  paths  of  the  light  rays.  Further,  when- 
ever a  ray  of  light  passes  from  free  space  into,  or 
through  a  medium  of  any  kind  or  density,  such  ray  is 
refracted,  or  bent  out  of  its  straight  course.  The  path 
of  such  a  ray  becomes  curved,  and  the  amount  of  re- 
fraction, or  curvature,  depends  upon  the  density  of  the 
medium  into  which  the  ray  passes  and  the  angle  at 
which  it  meets  the  surface.  This  is  a  fundamental  law 
of  physics:  upon  the  refractive  effects  of  different 
media  are  based  our  optical  instruments  and  experi- 
ments :  eye-glasses,  cameras,  microscopes,  telescopes ;  all 
depend  upon  the  refractive  effect  of  glass  upon  a  ray  of 
light.  It  is  certain,  therefore,  that  the  rays  of  light,  in 
passing  through  the  solar  envelope,  suffered  a  refrac- 
tion, or  bending,  of  some  kind  and  amount.  This  fact 
is  as  well  established  as  the  existence  of  the  sun  itself. 
The  sole  question  is  whether  this  refraction  was  suffi- 
cient in  amount  and  in  direction  to  account  for  the 
observed  displacements  of  the  star  images. 

This  possibility  of  accounting,  in  a  perfectly  normal 
way,  for  the  observed  light  deflections  has  been  dis- 
missed by  the  relativitist  in  a  few  words  as  a  matter 
scarcely  worth  mentioning. 

While  it  is  certain  that  the  rays  suffer  some  refraction 


Classical  Methods  241 

in  passing  through  the  solar  envelope,  it  is  claimed  by 
most  astro-physicists  that  the  effect  is  so  small  as  to  be 
negligible  in  comparison  with  the  observed  deflections. 
This  idea  is  so  firmly  fixed  that  the  possibility  of  ex- 
plaining any  portion  of  the  deflections  by  refraction 
was  dismissed  by  the  British  astronomers  in  their 
Report  with  a  scant  phrase  or  two.  The  entire  question 
depends  upon  the  possibility  of  the  solar  envelope 
having  a  density  large  enough  to  bend  a  ray  of  light 
by  the  required  amount,  and  this  in  turn  upon  what 
that  density  really  is. 

It  can  readily  be  shown  by  the  ordinary  formulas  of 
optics  that  a  lens  of  matter  of  a  density  of  about 
i/i4Oth  that  of  air  at  standard  pressure  and  tempera- 
ture would  deflect  a  ray  of  light  by  about  i" ',  the 
amount  observed  in  the  case  of  the  star  nearest  the  sun. 
And  this  is  the  density  that  the  astro-physicist  assumes 
to  be  necessary  to  account  for  the  observed  deflections. 
And,  of  course,  matter  of  such  density  cannot  exist  in 
the  immediate  vicinity  of  the  sun,  where  the  light  rays 
passed.  This  is  not  a  physical  impossibility;  there  is 
no  fundamental  law  of  physics  or  mechanics  that 
renders  such  a  density  an  impossibility;  rather  is  it  an 
observational  improbability,  made  so  by  other,  definite 
observations.  Such  matter,  if  it  existed,  would  refract, 
or  scatter,  the  rays  of  the  sun  itself  in  all  directions,  and 
the  sun  would  appear  surrounded  by  a  large,  brilliant 
halo,  or  corona.  There  is  a  mathematical  relation, 

16 


242    Gravitation  versus  Relativity 

depending  upon  physical  laws,  between  the  density  of 
the  matter  and  the  size  and  brilliancy  of  the  surrounding 
halo  of  the  sun.  The  actual  corona  of  the  sun,  as 
observed  and  photographed,  would  be  far  larger  and  far 
more  brilliant  than  it  really  is.  From  estimates  of  the 
brightness  of  the  solar  envelope  at  the  points  where 
the  rays  passed,  the  physicist  concludes  that  the  density 
of  the  envelope  cannot  be  greater  than  one  one-thou- 
sandth part  of  that  deemed  necessary  to  produce  the 
refraction. 

Now  all  this  reasoning  depends  upon  the  assumption 
that  the  density  of  the  solar  envelope  must  be  approxi- 
mately i/i40th  that  of  the  air  in  order  to  bend  the  ray 
by  the  requisite  amount.  If  this  assumption  be  greatly 
in  error,  then  the  whole  argument  fails.  But  the  ordi- 
nary formulas,  upon  which  this  assumption  is  based, 
for  refraction  due  to  a  lens,  or  to  the  atmosphere,  do 
not  apply  to  the  case  of  a  ray  of  light  passing  through 
the  solar  envelope,  as  did  the  rays  on  the  eclipse  plates. 
In  the  customary  optical  formulas  for  lenses,  the  re- 
fractive medium  is  assumed  to  be  of  uniform  density 
throughout,  whilst  in  the  cases  of  the  earth's  atmos- 
phere and  of  the  solar  envelope,  the  density  varies 
through  wide  limits,  decreasing  as  the  distance  from 
the  central  body  becomes  greater  and  greater.  Again 
the  ordinary  formula  for  refraction  through  the  earth's 
atmosphere  is  derived  upon  the  assumption  that  the 
angle  of  incidence  of  the  ray  remains  sensibly  constant 


Classical  Methods 


245 


while  the  ray  is  passing  from  the  highest  to  the  lowest 
strata  of  the  air.  This  condition,  however,  is  not 
filled  when  a  ray  passes  nearly  centrally  through  a 
globular  mass  of  varying  densities.  As  such  a  ray 
passes  from  the  outer  limits  of  the  mass  towards  the 
central  portion,  it  meets  concentric  layers  of  greater 
and  greater  densities  and  of  smaller  and  smaller  radii  of 
curvature.  The  angle  of  incidence  of  the  ray,  therefore, 
increases  as  the  ray  approaches  the  centre  of  the  mass 
and  tends  to  become  90°  at  the  point  of  nearest  ap- 
proach. At  this  point  the  ordinary  formula  fails  com- 
pletely, for  one  of  the  factors  increases  without  limit 
and  makes  the  refraction  apparently  enormous.  This 
is  shown  in  Figure  28,  where  A  is  a  ray  of  light  from 
the  sun  or  a  star  as  ordinarily  observed,  and  B,  a  ray 


Fig.  28.    Failure  of  Refraction  Formulas. 

passing  through  the  atmosphere  very  nearly  in  a  hori- 
zontal direction.  The  ray  A  meets  the  successive  layers 
of  the  atmosphere  at  nearly  the  same  angle,  but  the 
ray  B  meets  the  various  layers  at  angles  increasing  from 


244    Gravitation  versus  Relativity 

about  45°,  where  it  enters  the  atmosphere,  to  practically 
90°  where  it  grazes  the  surface  of  the  earth. 

Now  all  the  ordinary  formulas  and  theoretical  dis- 
cussions of  refraction  fit  rays  similar  to  A,  but  fail 
for  rays  similar  to  B.  But  even  for  the  A  rays  the 
theoretical  formulas  are  not  wholly  satisfactory.  The 
path  of  the  ray  depends  upon  the  way  in  which  the 
atmosphere  decreases  in  density  as  the  height  above 
the  surface  of  the  earth  increases,  and  practically  noth- 
ing is  known  as  to  the  law  of  the  decrease  of  density. 
Near  the  surface  of  the  earth  the  atmosphere  decreases 
in  density  very  rapidly,  but  at  higher  altitudes  it  seems 
to  decrease  much  more  slowly.  This  question  of  density 
is  involved  with  that  of  temperature,  and  the  de- 
crease of  temperature  with  altitude.  Various  formu- 
las have  been  derived  to  express  the  relation  between 
density,  temperature,  and  altitude,  but  without  com- 
plete success.  All  the  formulas  fail  to  fit  observed  con- 
ditions, except  very  near  the  surface  of  the  earth.  As 
higher  and  higher  altitudes  have  been  reached  by  small 
balloons  carrying  instruments,  the  discrepancies  of  the 
various  formulas  have  become  more  and  more  apparent. 

While  there  is,  thus,  no  satisfactory  theoretical 
formula  for  refraction,  there  are  tables  in  common  use 
which  give  the  refraction  of  a  ray  with  sufficient  ac- 
curacy for  ordinary  astronomical  purposes.  They  are 
extremely  accurate  for  rays  which  come  nearly  verti- 
cally through  the  atmosphere,  but  are  little  more  than 


Classical  Methods  245 

approximations  for  rays  which  reach  the  observer  in  a 
horizontal  direction,  like  those  from  the  setting  sun. 

Now  it  is  matter  of  common  observation  that  the 
atmospheric  refraction  of  a  horizontal  ray  is  about  35'. 
The  apparent  diameter  of  the  sun  is  slightly  less  than 
this,  so  that,  due  to  this  bending  of  the  light  rays,  the 
sun  is  visible  for  some  moments  after  it  has  really 
passed  below  the  horizon.  Thus  the  sun  rises  earlier 
and  sets  later  than  it  would,  were  the  earth's  atmosphere 
removed.  Now  the  total  bending  of  a  ray  of  light, 
passing  entirely  through  the  atmosphere  of  the  earth 
from  side  to  side  and  just  grazing  the  surface,  would  be 
double  the  above  amount,  or  approximately  4100".  The 
maximum  observed  deflection  on  the  Sobral  plates  is 
almost  exactly  i",  or  i/4iooth  that  of  the  earth's  at- 
mosphere upon  a  similar  ray.  As,  under  similar  condi- 
tions, the  amount  of  refraction  is  proportional  to  the 
density  of  the  medium,  it  would  appear  that,  if  the 
earth's  atmosphere  were  reduced  in  density  to  i/4iooth 
of  its  normal  amount,  it  would  still  refract  a  horizontal 
ray  of  light  by  i",  the  maximum  amount  of  the 
measured  star  deflections.  This  is  only  about  i/3Oth 
of  that  deemed  essential  by  British  astronomers. 

Newcomb  reinvestigated  the  theories  and  formulas 
of  atmospheric  refraction,  and  deduced  a  formula  for 
the  curvature  of  a  horizontal  ray,  as  it  passes  through 
the  earth's  atmosphere.  This  formula  of  Newcomb 
takes  into  account  changes  in  temperature  and  pressure, 


246    Gravitation  versus  Relativity 

and  gives  very  satisfactory  results  for  normal  condi- 
tions of  the  atmosphere:  further,  it  fits  abnormal  con- 
ditions and  gives  a  clear  explanation  of  mirages  and 
other  abnormal  phenomena. 

Now,  if  this  formula  of  Newcomb  be  applied  to  the 
case  of  a  ray  of  light  passing  through  the  solar 
envelope,  it  becomes  apparent,  if  the  formula  be  applica- 
ble, that  a  very  small  density  will  suffice  to  account  for 
a  refraction  of  i";  a  density  many  times  smaller  than 
has  hitherto  been  deemed  essential.  But  when  one  thus 
attempts  to  reason  by  analogy  from  conditions  on  the 
earth  to  conditions  near  the  sun,  and  to  apply  formulas 
derived  to  fit  conditions  on  the  earth  it  must  be  done 
with  full  reservations  as  to  its  validity.  The  conditions 
in  the  solar  envelope  are  so  radically  different  from 
anything  known  on  the  earth,  that  the  application  of  a 
formula,  which  gives  consistent  results  on  the  earth, 
may  lead  one  into  serious  complications.  But  this 
applies  to  the  formulas  used  by  the  astro-physicists, 
which  appear  to  show  that  a  high  density  is  necessary, 
as  well  as  to  the  formula  of  Newcomb,  which  indicates 
an  extremely  low  density. 

This  whole  question  of  refraction,  even  in  the  earth's 
atmosphere,  is  very  confused  and  complicated.  New- 
comb  in  1906  wrote,  "There  is,  perhaps,  no  branch  of 
practical  astronomy  on  which  so  much  has  been  written 
as  on  this  and  which  still  is  in  so  unsatisfactory  a 
state." 


Classical  Methods  247 

In  view  of  these  different,  often  conflicting,  for- 
mulas, with  all  the  complicated  and  largely  unknown 
conditions  in  the  solar  envelope,  it  is  certainly  not 
proved  that  cosmic  refraction  is  an  impossibility. 

While,  thus,  there  is  a  very  open  question  as  to  the 
amount  of  refraction  which  would  be  caused  by  a 
medium  of  varying  density,  there  is,  on  the  other  hand, 
practically  no  question  as  to  the  direction  in  which  the 
bending  would  take  place.  This  is  purely  a  matter  of 
geometry,  and  depends  upon  the  fundamental  law,  that 
the  incident  ray,  the  normal  to  the  surface,  and  the 
refracted  ray,  all  lie  in  the  same  plane.  Provided  solely, 
therefore,  that  each  concentric  infinitesimally  thin  layer 
is  of  uniform  density  and  of  a  geometric  shape,  the 
direction  in  which  the  ray  is  refracted  can  be  found  by 
geometrical  methods,  regardless  of  the  actual  density 
of  the  layer. 

A  spherical  shell  of  matter  is  symmetrical  with 
regard  to  the  centre;  the  normals  at  every  point  of  the 
surface  pass  through  the  centre,  and  thus  any  and 
every  ray  of  light  passing  through  such  a  surface  will 
be  refracted  radially.  From  whatever  point  such  a 
mass  of  matter  be  viewed,  a  ray  of  light  coming  through 
it  would  appear  bent  towards  the  centre.  Not  so,  how- 
ever, for  lenticular  masses,  or  for  masses  of  an 
ellipsoidal  shape.  In  an  ellipsoid  the  normals  from 
only  four  points  of  the  surface  pass  through  the  centre 
of  figure;  and  the  refraction,  in  general,  will  be  non- 


248    Gravitation  versus  Relativity 

radial.  Only  when  the  eye  of  the  observer  is  in  partic- 
ular positions  with  respect  to  the  axes  of  figure  and 
when  the  rays  of  light  come  through  particular  parts 
of  the  surface  will  the  refraction  be  radial,  or  directed 
towards  the  centre  of  figure.  But,  if  the  position  of 
the  observer,  relative  to  the  axes  be  known,  and  also 
the  point  in  which  the  ray  cuts  the  surface,  then  from 
formulas  of  geometry  can  be  determined  the  plane  in 
which  the  refraction  takes  place,  and  thence  the  de- 
parture of  the  deflection  from  radiality.  In  the  general 
case  of  an  ellipsoid  of  matter  the  formulas  become 
rather  long  and  intricate. 

In  the  case  of  the  photographs  taken  at  Sobral  dur- 
ing the  eclipse  of  May  29,  1919,  however,  an  approxi- 
mate solution  may  be  obtained  with  great  simplicity. 
For,  assuming  the  solar  envelope  to  be  an  ellipsoid  of 
revolution  with  its  axis  coinciding  with  that  of  the 
sun,  the  axis  of  figure  would  be  practically  at  right 
angles  to  the  line  of  sight.  On  June  3rd  this  would 
have  been  strictly  true,  and  on  the  day  of  the  eclipse 
the  axis  was  tilted  towards  the  earth  at  an  angle  of  only 
i  ° ;  an  angle  so  small  that  its  effects  upon  the  quanti- 
ties can  be  neglected  in  an  approximate  solution. 

But  in  order  to  apply  any  formulas  to  the  solar  en- 
velope, some  assumption  must  be  made  as  to  its  general 
size  and  shape.  For  the  purpose  of  illustration  the 
major  axis  of  the  spheroid  may  be  taken  as  fifteen  times 
the  radius  of  the  sun  and  the  ellipticity  as  0.4.  The 


Classical  Methods  249 

various  values  of  the  departures  from  radiality,  as  cal- 
culated upon  these  assumptions,  are  given  in  the  follow- 
ing table,  where  for  comparison  are  also  given  the 
actual  observed  values,  as  determined  from  the  Report 
of  the  British  astronomers  and  as  shown  on  the  vari- 
ous diagrams  of  the  eclipse  plates  on  page  215. 

TABLE  V. 

Angular  Departures  from  Radiality 

Star  No.      Observed       Calculated 


3  ~    3° 


2  4-10°  4-  26° 

4  +1°  -4° 

5  -    4°  -25° 

6  -  16°  -  13° 

10  -  28°  -  19° 

11  4-36°  +24° 

These  calculated  departures  from  radiality  agree  in 
a  striking  way  with  the  observed  values.  It  will  be 
noted  at  once  that,  with  the  exception  of  Star  4,  all 
the  calculated  departures  have  the  same  sign  as  the 
observed.  The  agreement  between  the  calculated  and 
the  observed  departures  from  radiality  is  very  good 
for  the  four  stars,  Nos.  3,  6,  10,  and  n;  the  calcu- 
lated departure  for  two  stars,  No.  2,  and  5,  however, 
are  very  much  larger  than  the  observed. 

In  the  above  calculations  the  dimensions  of  the  solar 
ellipsoid  were  arbitrarily  assumed.  In  order  to  test 
the  general  conclusions,  the  calculations  were  repeated 


250    Gravitation  versus  Relativity 

with  various  ellipsoids,  the  major  axes  of  which  varied 
from  10  radii  of  the  sun  to  18  radii,  and  the  ellipticities 
from  0.2  to  0.5.  In  every  case  the  results  were  the 
same,  the  calculated  departures  from  radiality  showed 
a  strong  general  resemblance  to  the  observed  depar- 
tures. Changes  in  the  size  of  the  ellipsoid  made  very 
small  relative  changes  in  the  departures,  but  changes 
in  the  ellipticity  produced  marked  changes  in  the  re- 
sults. As  the  ellipticity  is  increased,  the  calculated  de- 
partures in  the  cases  of  the  outer  star  images  agree 
more  closely  with  the  observed  values;  but  in  the  case 
of  the  inner  stars,  especially  Nos.  2  and  5,  smaller 
ellipticities  fit  the  observations  very  much  better.  Fur- 
ther, in  all  these  calculations  the  axis  of  revolution  of 
the  mass  of  matter  was  taken,  for  simplicity,  as  being 
perpendicular  to  the  line  of  sight.  If  this  axis  of 
figure  were  tilted  towards  or  away  from  the  observer, 
then  these  calculated  refractive  angles  would  change; 
and  such  tilting  would  affect  the  different  stars  differ- 
ently. 

Now  it  is  not  a  difficult  matter  to  find  by  the  methods 
of  Least  Squares  the  general  shape  and  position  in 
space  of  that  ellipsoid,  which  would  so  refract  the 
rays  from  the  different  stars,  as  to  most  nearly  repre- 
sent the  actual  observed  deflections  (in  direction  only). 
The  best  solution  shows  the  axis  to  be  tilted  towards  the 
observer  by  an  amount  slightly  over  2°.  The  results 
for  the  individual  stars  are  given  in  the  following 


Classical  Methods  251 

table,  where  they  are  compared  with  the  observed  de- 
flections and  with  those  of  Einstein,  which,  of  course, 
are  zero  in  every  case. 

TABLE  VI 
Computed  Departures  from  Radiality 

Star  No.      Observed  Computed 

Poor          Einstein 


3 

-    3° 

-    6° 

0 

2 

+  10° 

+    5° 

0 

4 

+     i° 

-    5° 

0 

5 

-    4° 

-  18° 

o 

6 

-  16° 

-  11° 

o 

10 

-28° 

-  18° 

o 

ii 

+  36° 

+  33° 

o 

It  needs  only  a  glance  at  the  figures  to  show  how  very 
much  better  the  hypothesis  of  refraction  represents 
the  observed  quantities  than  does  the  hypothesis  of 
relativity.  This  comparison  of  results  can  also  be  made 
by  the  ordinary  method  of  taking  the  sum  of  the 
squares  of  the  residuals,  which  method  gives, 

Relativity  theory,  2,489 

Refraction  hypothesis,  410 

The  great  reduction  shown  by  the  refraction  hypo- 
thesis indicates  clearly  its  superiority  over  that  of 
Einstein.  This  is  also  shown  in  the  following  diagram, 
which  represents  the  eclipse  field,  and  shows  for  each 
star  the  observed  deflection,  the  theoretical  Einstein 


252    Gravitation  versus  Relativity 

effect,  and  the  computed  refraction  effect  in  direction 
only. 

Further  it  was  found  that  the  ellipsoid  which  gave 
the  refractive  effect  most  nearly  representing  the  ob- 


n 


SCALE    I"  •     i 
OBSERVED  DEFLECTIQH 
EINSTEIN  DEFLECTION* 
RE.FRACTJON  - 


t 


Fig.  29.    Comparison  of  the  Einstein  and  Refraction  Effects. 

served  deflections  had  its  axis  tilted  in  the  line  of  sight. 
The  principal  plane  of  the  ellipsoid  is,  of  course,  at  right 
angles  to  this  axis,  and  the  position  of  this  plane,  in 
reference  to  the  ecliptic,  is  thus  found  to  be 


Classical  Methods  253 


Longitude  of  the  Node  44 

Inclination  7 


Now  in  the  discussion  of  the  motions  of  the  planets 
and  of  the  discordance  in  the  motion  of  the  perihelion 
of  Mercury,  the  position  was  found  for  a  ring,  or  an 
ellipsoid  of  matter,  within  the  orbit  of  Mercury,  which 
would  best  account  for  the  various  motions.  This 
determination  was 

Longitude  of  the  Node  36° 

Inclination  7°.  5 

Newcomb,  in  his  investigation  published  in  1895,  gave 
for  the  values  of  these  quantities 

Longitude  of  the  Node  48° 

Inclination  9° 

This  is  a  striking  fact.  Two  radically  different  in- 
vestigations, one  on  the  motions  of  the  planets,  the 
other  on  the  deflections  of  light  rays,  both  lead  to  prac- 
tically the  same  ellipsoid  of  matter. 

These  results  indicate,  at  least,  the  possibility  of 
explaining  the  observed  light  deflections,  if  such  de- 
flections be  real,  by  the  refraction  of  the  rays  during 
their  passage  through  the  solar  envelope,  the  shape  of 
which  is  generally  that  of  an  oblate  spheroid. 


CHAPTER  IX 

CONCLUSIONS 

THE  astronomical  evidence,  cited  by  Einstein  as 
complete  and  satisfactory  proof  of  the  relativity  theory, 
fails  to  support  his  hypothesis.  His  hypotheses  and 
formulas  are  neither  necessary  nor  sufficient  to  explain 
the  observed  phenomena.  They  are  not  sufficient,  for 
they  account  for  only  one  of  the  numerous  discordances 
in  planetary  motions,  for  only  a  portion  of  the  sup- 
posed light  deflections :  they  are  not  necessary,  for  all 
the  discordances  in  the  motions  of  the  planets,  includ- 
ing that  of  Mercury,  can  readily  be  accounted  for  by 
simple  gravitational  methods,  and  the  light  deflections, 
if  real,  can  be  equally  well  explained  on  other  grounds. 

A  motion  of  the  perihelion  of  Mercury,  similar  and 
approximately  equal  to  that  actually  observed,  can  be 
explained  by  the  Einstein  hypothesis.  But  this  hypo- 
thesis fails  completely  to  explain  other  motions  of 
Mercury  and  similar  motions  in  other  planets,  it  causes 
new  and  inexplicable  discordances  in  the  motion  of 
Venus.  On  the  other  hand,  all  the  observed  motions 
of  both  Mercury  and  Venus  can  readily  be  explained 

254 


Conclusions  255 

by  the  action,  under  the  Newtonian  law  of  gravitation, 
of  masses  of  matter,  known  to  exist.  And  such  ex- 
planation is  based  upon  formulas  and  methods,  known 
and  used  for  well  over  a  century  to  account  for  similar 
motions  in  other  portions  of  the  solar  system. 

Deflections  of  light  rays,  similar  to  those  reported 
at  Sobral,  can  be  explained  by  the  relativity  theory. 
This  hypothesis  can  account,  very  approximately,  for 
the  amount  of  the  supposed  deflections,  but  it  fails 
completely  to  account  for  the  directions  in  which  such 
deflections  occurred.  Refraction  by  the  cosmic  matter, 
through  which  the  rays  are  known  to  have  passed,  will 
account  fairly  well  for  the  observed  directions,  but  en- 
counters very  serious  difficulties,  in  accounting  for  the 
amounts  of  the  deflections,  as  reported. 

But  for  the  true  relativitist  the  pathway  through  all 
the  difficulties  of  conflicting  evidence  is  smooth  and 
clear;  for  does  not  everything  depend  upon  the  ob- 
server? Nothing  is  absolute,  everything  is  relative; 
the  statue  is  golden  for  one  observer  and  silver  to  the 
other.  To  the  relativitist  the  motion  of  the  perihelion 
of  Mercury,  of  course,  is  real  and  is  exactly  the  43" 
required  by  the  Einstein  hypothesis,  but  the  other 
motions  do  not  exist,  they  are  mere  accidental  errors. 
It  makes  no  difference  that  all  these  various  motions 
result  from  the  same  investigations,  that  both  Lever- 
rier  and  Newcomb  show  that  the  motion  of  the  peri- 
helion is  not  independent,  that  it  must  be  accompanied 


256    Gravitation  versus  Relativity 

by  and  depend  upon  other  motions.  These  other  mo- 
tions cannot  be  explained  by  relativity,  and,  therefore, 
they  do  not  exist,  they  have  not  been  "sufficiently 
attested/'*  Thirty-three  photographic  plates,  taken 
during  the  eclipse  of  1919,  show  star  images;  of  these 
thirty-three,  seven  only  give  results  even  approximating 
towards  the  Einstein  predictions.  And  to  make  even 
these  seven  fit  the  hypothesis,  the  relativitist  is  forced 
to  invoke  the  aid  of  the  sun  to  distort  the  camera  in 
a  particular  way  and  by  just  the  right  amount! 

The  explanation  of  the  old  fashioned  astronomer, 
that  the  motion  of  Mercury  may  be  due  to  masses  of 
matter,  which  have  been  seen  and  photographed  many 
times,  is  dismissed  by  the  relativitist  as  having  little 
probability  and  as  having  been  devised  solely  for  the 
purpose.  The  corona  of  the  sun  has  been  known  from 
pre-historic  times,  the  zodiacal  light  for  many  years, 
and  meteors  have  fallen  to  the  earth  in  all  ages.  That 
an  elliptical  shaped  central  body  would  cause  a  perihe- 
lial  motion,  was  shown  by  Walmsley  in  1758,  and  by 
him  used  to  explain  the  motions  of  Jupiter's  satellites. 
Was  this  devised  solely  to  explain  the  motion  of  Mer- 
cury? Did  Walmsley  devise  a  method  for  the  sole 
purpose  of  explaining  something  of  which  he  was  en- 
tirely ignorant,  and  which  was  not  discovered  until 
nearly  a  century  after  his  death?  The  corona,  the 
zodiacal  light,  meteors,  are  these  fictions  of  the  imagin- 

*  Albert  Einstein:  letter  of  July  30,  1921. 


Conclusions  257 

ation?  Were  these  devised  by  the  deluded  followers 
of  Newton  solely  to  explain  the  motions  of  Mercury? 
The  relativity  theory  may  be  true,  but  no  substantial 
experimental  proofs  have  yet  been  submitted  by  any  of 
its  adherents. 

17 


APPENDICES 


259 


APPENDIX  I 

THE  MICHELSON-MORLEY  EXPERIMENT  ON 
ETHER-DRIFT 

THE  Michelson-Morley  experiment  forms  the  basis 
of  the  relativity  theory :  Einstein  calls  it  decisive.  If  it 
should  be  shown  that  this  experiment  is  not  decisive, 
that  the  negative  results  obtained  were  due  to  instru- 
mental errors  or  to  some  peculiar  conditions  under 
which  the  experiments  were  conducted;  if  it  should 
develop  that  there  is  a  measurable  ether-drift,  then 
the  entire  fabric  of  the  relativity  theory  would  collapse 
like  a  house  of  cards.  For  this  reason  the  repetitions 
of  the  Michelson-Morley  experiment  recently  made  at 
Cleveland  and  at  Mount  Wilson  are  of  especial  im- 
portance: they  indicate  that  the  original  experiment 
was  not  decisive,  and  that  there  may  be  a  measurable 
ether-drift. 

Many  years  ago  it  was  suggested  that  the  negative 
result  of  the  Michelson-Morley  experiment  might  be 
due  to  the  earth  dragging  the  ether,  in  its  immediate 
vicinity,  along  with  it:  that  the  ether  in  the  room,  in 
which  the  experiment  was  made,  was  entrapped  and 

261 


262  Appendix 

moved  with  the  room.  A  motor-boat,  a  steamship, 
moving  through  still  water  drags  the  particles  of  water, 
in  immediate  contact  with  its  sides,  along  with  it.  If 
one  looks  directly  down  from  the  deck  of  a  moving 
vessel,  one  will  see  the  particles  of  water  apparently 
cling  to  the  sides  of  the  boat  and  move  forward  with 
the  boat;  particles  an  inch  or  two  from  the  surface 
cling  less  tenaciously  and  are  slowly  passed;  particles 
a  foot  or  two  from  the  sides  show  no  frictional  effect 
and  are  left  at  rest  by  the  passing  vessel.  To  measure 
the  true  speed  of  the  vessel  through  the  water,  one 
would  have  to  consider  the  motion  of  the  hull  relative 
to  water  particles  some  considerable  distance  away  from 
the  sides  of  the  boat  This  effect  of  dragging  water  is 
the  well  known  phenomenon  of  "skin  friction,"  which 
plays  such  an  important  part  in  the  design  of  all  vessels. 

The  attempts  of  Michelson  and  Morley  to  measure, 
in  the  basement  of  buildings  and  at  low  altitudes,  the 
motion  of  the  earth  through  the  ether  might,  not  in- 
aptly, be  compared  to  the  attempts  of  some  minute 
beings,  living  in  a  small  rust-pit  on  the  side  of  the 
Leviathan,  to  measure  the  speed  of  that  immense  ves- 
sel through  the  waters  by  experiments  in  the  thin  film 
of  water,  contained  entirely  within  the  hollow  in  which 
they  lived. 

In  the  years  1891-1897,  Sir  Oliver  Lodge  tested  this 
idea  of  skin  friction  between  moving  bodies  and  the 
ether,  and  attempted  to  measure  the  amount  of  such 


The  Michelson-Morley  Experiment  263 

friction,  if  any  there  be.  He  devised  an  elaborate  ap- 
paratus, by  which  he  could  test  whether  the  ether  con- 
tained between  two  parallel  steel  plates  was  dragged 
along  by  the  plates,  when  they  were  whirled  at  high 
speed.  His  experiments  showed  that  the  ether  between 
such  discs,  or  plates  was  not  dragged  sufficiently  to 
change  the  velocity  of  light  by  so  much  as  the  i/ioooth 
part  of  the  velocity  of  the  plates.  And  he  concluded 
from  this  experiment  that  the  viscosity,  or  fluid  friction 
of  the  ether  is  zero.  In  considering  this  result  it  must 
be  remembered  that  the  discs,  or  plates  were  only  some 
three  feet  in  diameter,  and  were  placed  about  one  inch 
apart.  The  earth  is  some  forty-two  million  feet  in 
diameter.  Thus  this  attempt  of  Sir  Oliver  Lodge  to 
detect  possible  skin  friction  of  the  earth  is  not  radically 
different  from  an  attempt  of  a  naval  architect  to  find 
the  skin  friction  of  the  Leviathan,  after  several  months 
in  service,  from  tests  made  on  a  plate  of  highly  polished 
metal  one  inch  in  diameter. 

Now  the  possibility  of  skin  friction  between  the  earth 
and  the  ether  can  be  tested  by  repeating  the  Michelson- 
Morley  experiment  at  different  distances  above  the 
earth's  surface.  An  accurate  test,  of  course,  can  only 
be  made  in  the  higher  regions  of  the  atmosphere,  clear 
above  the  tops  of  the  highest  mountains.  This  is  im- 
possible, but  it  is  possible  to  utilize  high  altitude  stations 
and  compare  the  results  with  those  obtained  at  ordinary 
levels.  This  has  been  done  by  Professor  Dayton  C. 


264  Appendix 

Miller  at  the  Mount  Wilson  Observatory,  at  an  alti- 
tude of  some  6,000  feet.  He  there  made  the  experi- 
ment with  the  original  apparatus  used  by  Morley 
and  Miller,  and  repeated  it  with  improved  instruments. 
He  summarized  his  findings  in  the  following  words  :* 

"The  suggestion  was  then  made  that  the  earth  drags 
the  ether,  and  while  there  is  no  'driff  at  the  surface  of 
the  earth,  it  might  be  perceptible  at  an  elevation  above 
the  general  surface.  The  experiment  was  again  performed 
by  the  present  author  at  the  Mount  Wilson  Observatory 
in  March  and  April,  1921,  where  the  elevation  is  nearly 
6,000  feet.  The  results  indicated  an  effect  such  as  would 
be  produced  by  a  true  ether-drift,  of  about  one  tenth  of 
the  expected  amount,  but  there  was  also  present  a  periodic 
effect  of  half  the  frequency  which  could  not  be  explained. 
The  interferometer  had  been  mounted  on  a  steel  base  and 
in  order  to  eliminate  the  possibility  of  magnetic  disturb- 
ance, a  new  apparatus  with  concrete  base  and  with  alum- 
inum supports  for  the  mirrors  was  constructed.  Obser- 
vations were  made  in  November  and  December,  1921,  the 
results  being  substantially  the  same  as  in  April.  Before 
any  conclusions  can  be  drawn,  it  is  necessary  to  deter- 
mine the  cause  of  the  unexplained  disturbance." 

These  experiments  of  Professor  Miller  are  not  con- 
clusive, but  they  appear  to  indicate  that  the  ether  is 
dragged  along  by  the  rough  surface  of  the  earth,  and 

*  Science,  No.  1427 :    May  5,  1922. 


The  Michelson-Morley  Experiment  265 

that  the  true  drift  might  be  measured  if  one  could 
attain  a  sufficient  height  above  the  surface  of  the  earth. 
If  there  be  an  ether-drift,  as  these  experiments  indicate, 
then  the  entire  structure  of  the  relativity  theory  is  ren- 
dered worthless.  But,  whether  there  ultimately  prove 
to  be  a  measurable  ether-drift  at  high  altitudes  or  not, 
this  cautious  statement  of  Professor  Miller  embodies 
the  true  scientific  spirit,  and  is  in  marked  contrast  to 
the  statements  and  assertions  of  the  relativitists. 


APPENDIX  II 

EINSTEIN    AND    THE    FIZEAU    EXPERIMENT 

THE  treatment  of  the  Fizeau  experiment  by  Einstein 
requires  a  few  words  of  explanation.  He  gives  two 
equations  as  follows  : 

W=v  +  w     ......     (A). 

W=*-±*  .  (B), 


in  which  v  is  the  velocity  of  the  water  in  the  tube,  w 
the  velocity  of  light  in  a  motionless  fluid,  and  W  the 
velocity  of  light  relative  to  the  tube. 

He  states  that  equations  A  and  B  represent  the  rela- 
tions between  these  quantities,  A  according  to  the  or- 
dinary theories  of  classical  mechanics,  and  B  according 
to  the  relativity  theory.  He  then  shows  that  the  rela- 
tivity equation  B  more  nearly  represents  the  results 
of  Fizeau's  observations:  "Experiment  decides  in 
favour  of  equation  (B)  derived  from  the  theory  of 
relativity,  and  the  agreement  is,  indeed,  very  ex- 
act' '(48). 

Equation  (A),  however,  is  not  an  equation  of  classi- 

266 


The  Fizeau  Experiment         267 

cal  optics;  it  is  found  nowheres,  except  in  Einstein;  it 
has  nothing  whatsoever  to  do  with  Fizeau's  experiment. 
As  it  stands  it  is  a  mere  statement  that  the  velocity  of 
light  in  the  moving  water  is  equal  to  the  sum  of  the 
velocities  of  light  in  air  and  of  the  water  in  the  tube. 
This  has  never  been  claimed.  Every  formula,  hereto- 
fore used,  has  involved  a  quantity  that  Einstein  omits, 
namely,  the  index  of  refraction  of  water. 

Further,  the  results  obtained  from  equation  (B)  are 
not  identical  with  the  observational  results  of  Fizeau. 
In  order  to  bring  equation  (B)  into  accord  with  the 
results  of  Fizeau,  Einstein  is  obliged  to  make  approxi- 
mations, or  to  neglect  certain  terms  of  his  own  for- 
mula. By  means  of  such  approximations,  he  finally  puts 
his  equation  (B)  in  the  form: 


where  n  is  the  index  of  refraction  of  water,  equal  to 
the  ratio  c/w.  And  this  equation  is  identical  with 
Fizeau's. 

Thus  in  applying  his  "crucial  test,"  Einstein  sets  up 
and  knocks  down  an  equation  never  before  heard  of, 
an  equation  having  no  relevancy  to  the  observations 
discussed,  and  then  adjusts  his  own  equation,  by  a 
system  of  approximations,  to  fit  the  observations. 


APPENDIX  III 

THE   MATHEMATICS  OF    RELATIVITY 

THE  entire  relativity  theory  is  based  upon  certain 
assumptions,  or  postulates,  from  which  were  derived 
mathematically  all  the  complicated  formulas  and  con- 
clusions of  Einstein.  It  has  always  been  taken  for 
granted  that  the  mathematics  of  relativity  were  correct; 
that  the  conclusions  followed  logically  and  inevitably 
from  the  fundamental  premises  or  assumptions.  Now 
this  very  point  has  lately  been  investigated  by  eminent 
French  mathematicians,  especially  by  Painleve,*  who 
has  shown  that  a  number  of  different  formulas  can  be 
derived  in  the  manner  of  Einstein,  that  many  different 
and  inconsistent  conclusions  can  be  drawn  from  the 
fundamental  premises  of  relativity. 

From  his  formulas  Einstein  drew  certain  conclusions 
regarding  the  behavior  of  clocks  and  of  measuring 
rods,  when  in  motion.  Painleve  has  shown  that  the 
Einstein  formulas  are  not  the  only  formulas  to  be  de- 
rived from  the  premises,  that  there  is  an  infinity  of 

*  Classical  Mechanics  and  the  Theory  of  Relativity,  by  P. 
Painleve.  Science  Abstracts:  Section  A— Physics:  Vol.  25,  Part 
3,  page  170.  March  31,  1922. 

268 


The  Mathematics  of  Relativity  269 

other  possible  formulas.  One  of  these  other  possible 
formulas  leads  to  the  ordinary  results  of  Euclidean 
space  and  to  the  constancy  of  rigid  bodies.  Other 
possible  formulas  lead  to  the  conclusion  that  bodies  ex- 
pand instead  of  contracting,  still  others  that  they  ex- 
pand at  right  angles  to  the  direction  of  motion. 

The  conclusions  of  Einstein  appear  to  Painleve  to 
be  audacious  conjectures  and  not  the  inevitable  conse- 
quences of  the  premises:  he  concludes  that  it  is  pure 
imagination  to  pretend  to  draw  conclusions  such  as  Ein- 
stein does.  He  believes  that  a  number  of  Einstein's 
formulas  will  blend  with  classical  science,  but  that 
some  of  the  more  startling  consequences  of  the  theory 
will  not  finally  survive. 


APPENDIX  IV 

THE  DISPLACEMENT  OF  SOLAR  LINES  AND  RELATIVITY 

EINSTEIN  has  claimed  that  the  observations  of  Grebe 
and  Bachem  at  Bonn  on  the  cyanogen  lines  in  the  solar 
spectrum  place  the  reality  of  the  relativity  displacement 
almost  beyond  doubt,  and  in  these  observations  he  see? 
clear  experimental  confirmation  of  his  entire  theory. 
It  has  been  noted,  however,  in  Chapter  II  that  the  bands 
or  lines  of  the  solar  spectrum  are  subject  to  displace- 
ments due  to  other  causes,  to  motions  of  the  earth  and 
sun,  to  motions  of  the  solar  atmosphere,  and  to  differ- 
ences of  pressure.  These  displacements  may  be  much 
larger  than  the  predicted  Einstein  effect.  Thus,  the 
relativity,  or  Einstein,  shift  is  not  a  clear-cut  effect 
which  can  be  directly  measured ;  it  must  be  disen- 
tangled, if  it  exists,  from  several  similar,  overlapping, 
and  even  larger  effects. 

In  the  Annual  Report  of  the  Director  of  the  Mount 
Wilson  Observatory  of  California  is  to  be  found  a  sum- 
mary of  the  observations  upon  the  cyanogen  lines,  made 
by  various  observers,  each  of  whom  claims  to  have 

270 


Solar  Lines  and  Relativity       271 

proved  the  existence  of  the  Einstein  effect.    From  this 
summary,  the  following  facts  appear: 

PEROT  applied  corrections  for  downward  movement  of 
the  solar  atmosphere  and  for  negative  pressure 
shift  (approximately  equal  to  the  Einstein  shift), 
and  when  thus  corrected,  his  results  agreed  with 
the  Einstein  prediction. 

BIRGE  applied  a  correction  for  an  upward  movement  of 
the  atmosphere,  but  no  pressure  shift,  and  when 
thus  corrected,  his  results  agreed  with  the  Ein- 
stein prediction. 

GREBE  and  BACHEM  assumed  neither  upward  nor 
downward  movement  of  the  atmosphere  and  no 
pressure  shift,  but  applied  a  correction  for  a 
supposed  asymmetry  of  the  arc-lines,  and,  when 
thus  corrected,  their  results  agreed  with  the 
Einstein  prediction. 

Had  these  three  observers  applied  the  same  correc- 
tions, in  the  same  way,  it  is  perfectly  clear  that  their 
final  results  would  have  been  very  discordant,  and  that 
two  sets  of  results,  at  least,  would  have  differed  radi- 
cally from  the  predicted  Einstein  effect.  As  a  matter 
of  fact,  these  three  sets  of  observations,  taken  together, 
do  not  show  the  slightest  trace  of  the  relativity  effect; 
they  are  radically  discordant  and  can  only  be  made  to 
show  the  desired  result  by  arbitrary  and  contradictory 
corrections. 


272  Appendix 

Mr.  St.  John,  of  the  Mount  Wilson  Observatory, 
sums  up  the  various  observations,  in  the  Annual  Re- 
port of  the  Director,  in  the  following  words  :* 

"Owing  to  the  different  and  even  inconsistent  correc- 
tions applied  to  the  observed  sun-arc  displacements,  the 
resulting  approximate  agreement  with  the  deductions  from 
the  Einstein  theory  fails  to  carry  conviction/' 

This  statement  is  certainly  conservative! 

*  Carnegie  Institution  of  Washington.  Annual  Report  of  the 
Director  of  the  Mount  Wilson  Observatory.  Year  Book,  No.  20, 
for  the  year  1921,  p.  244. 


INDEX 


Action,  109 

Addition  of  velocities,  28 
AMBRONNE,  figure  of  the  sun,  98 
Approximation,  79,  93,  95,  101, 

107,  123,  129,  149,  177,  179 
ARISTOTLE,  109 
Assumptions,  classical,  31 

of  Einstein,  34,  44,  268 
Atmosphere,  74,  152,  205 

resistance  of ,  in 
Attraction,  69,  80 

differential,  126 

of  irregular  bodies,  87 

of  particles,  84 

of  shells,  86 

of  spheres,  86 

of  spheroids,  88 
AUWERS,  figure  of  the  sun,  97 
Axes,  24,  99 


B 


BACHEM,  shift  of  spectral  lines, 

59,  270 

BIRGE,  shift  of  spectral  lines,  271 
Bodies,  problem  of  two,  118 

problem  of  three,  122 
Bonn,  59 
Book,  of  Einstein,  19 

of  Leverrier,  141,  161 

of  Newcomb,  167,  184 
BOUGUER,  transit  of  Mercury, 

163 
Brazil,  66,  198,  201 


Calculus,  118 

CASSINIS,   transit  of  Mercury, 
163 


Centre  of  gravity,  114 
Chaldeans,  76 
Circle,  47 
Clock,  35 
Ccelostat,  199,  209 
Comets,  1 02 
Conic  sections,  115 
Coordinate  planes,  24 

systems,  24,  39,  44,  46 

time,  43 
Coordinates,  24,  37,  39,  205 

four  dimensional,  41 
COPERNICUS,  73 

system  of,  75 
CROMMELIN,  198 


DAVIDSON,  198 

Deferent,  72 

Density  of  solar  envelope,  235, 

238,  241 
Discrepancies,    173,    189,    191, 

218,  225,  249 

Displacement,  of  spectral  lines, 
49,  56,  270 

of  stars,  65,  212,  218,  249,  255 
Distance,  31,  39 
DYSON,  198 

£ 

Eccentricity,  120,  161 
Eclipse,    observations,    53,    64, 
Chapter  VII 

plates,  Chapter  VII,  239 

measurement  of,  204 
Ecliptic,  105,  155,  252 
EDDINGTON,  198 
EINSTEIN,     evidence 

Chapters  VI,  VII,  VIII 

evidence  for,  Chapter  II 

theory  of,  Chapter  I 


273 


274 


Index 


Electrodynamics,  56 
Elements  of  orbit,  119 
Ellipse,  77,  88,  113 
Ellipsoid,  88 
Ellipticity,  of  planets,  94,  230 

of  solar  envelope,  250 

of  sun,  95,  177,  196,  229 
Epicycle,  70,  132,  139 
Equation,  for  distance,  40 

personal,  99 

of  transformation,  23,  25,  32, 

38,  48 

Equivalence,  principle  of,  46 
Ether,  7 

drift,  9,  19,  261 

properties  of,  8,  263 
EUCLID,  geometry  of,  39,  47 
EULER,  satellites  of  Jupiter,  49, 

230,  256 
EVERSHED,  shift  of  spectral  lines, 

59  . 

Experiment,  crucial,  56 
decisive,  9,  20,  30 
of  Fizeau,  53,  266 
of  Michelson-Morley,  Chapter 

I,  261 
-optical,  3,  7,  20 

F 

PERREL,  138 

FITZGERALD,  contraction  theory, 

17,33 

FIZEAU,  experiment  of,  53,  266 
Force,  108 

Formulas,  for  addition  of  veloci- 
ties, 28 

for  effect  of  motion,  32,  33 

of  attraction,  90,  114,  182 

of  Newton,  60 

of  refraction,  241 

of  relativity,  268 
Four  dimensions,  37 
Fourier's  series,  133 
Frame  of  reference,  23,  45 


GALILEO,  laws  of  motion,  109 

sun  spots,  95 

work  of,  79 
GASSENDI,   account   of   transit, 

158 


Gegenschein,  181 
Geometry,  37,  47,  50,  115 
Gravitation,   46,    50 

law  of,  43,  Chapter  III,  114, 

182 
Gravitational  field,  46,  49 

of  sun,  64 
GREBE,  shift  of  spectral  lines, 

59,  270 
Greenwich,  97,  199 


HALL,  hypothesis  of,   182,   193. 

196 
HALLEY,  83 

tables  of  Mercury,  153 
Heliometer,  97 
HILL,  agreement  with  Leverrier, 

167 
HIPPARCHUS,  work  of,   69,   75, 

132, 139 
Hyperbola,  117 


Imaginary,  symbol  of  an,  43 
Inclination,  119,  253 
Independence  of  space  and  time, 

3,  30 

Interferometer,  16 
Interval,  3,  36 

absolute,  42 

expression  for,  41 

imaginary,  43 
Invariant,  41 


Jupiter,   discovery  of  satellites 

of,  80 

motion  of  satellites  of,  94,  230, 
256 


KEPLER,  73 
laws  of,  77,  119 
tables  of  Mercury,  153 


Index 


275 


LA  CAILLE,  transit  of  Mercury, 
163 

LALANDE,  tables  of  Mercury,  153 

LA  PLACE,  existence  of  perturba- 
tions, 145 
motions  of  planets,  101 

Laws  of  nature,  25 

Lens,  199,  204,  210,  231,  239 

LESCARBAULT,  discovery  of  Vul- 
can, 165 

LEVERRIER,  motions  of  Mercury, 

61,  141,  153 

motions  of  planets,  101 

stability  of  solar  system,  147 
Lick  Observatory,  201 
Light,  curvature  of,  64 

deflection  of,  212 

emission  theory  of,  7 

interference  of,  14 

nature  of,  7 

velocity  of,  10,  27,  50 

wave  theory  of,  8 
Limiting  velocity,  33 
Line  of  Nodes,  120 
DE  LISLE,  transit  of  Mercury, 

163 

LODGE,  Fizeau's  experiment,  55 
viscosity  of  the  ether,  263 

Logarithms,  79 

Longitude,  of  the  Node,  121,  253 
of  perihelion,  120 

LORENTZ,  contraction  theory  of, 

17.33 

electrodynamics,  56 
Fizeau's  experiment,  55 
transformation  equations,  32, 
38,48 

M 

Major  axis,  60,  117,  120 
Mars,  174,  191,  234 
Mass  of  planets,  163,  170,  172 
Matter,  47,  55,   179,  196,  227, 

240,  256 
MAXWELL,  electrodynamics  of, 

56 

Mechanics,    celestial,    96,    102, 
148,  230,  233 

classical,  60,  62 
Meteors,  106,  256 


MICHELSON,      experiment      of, 

Chapter  I,  261 
Milky  Way,  105 
MILLER,  16,  263,  264 
MINKOWSKI,  expression  for  in- 
terval, 41 

space-time  relation,  38,  45 
Mirage,  246 
Mirror,  distortion  of,  202,  220, 

225 

Moon,  80,  230 
MORLEY,  experiment  of,  Chapter 

I,  261 
Motion,  4,  108 

accelerated,  46 

amplitude  of,  130 

circular,  72 

harmonic,  133 

laws  of,  26,  79,  1 08 

of  the  earth,  4,  9,  234 

of  Mercury,  60,  141,  Chapter 

V,  234,  254 

of  planets,  45,  77,  93,  Chapter 

VI,  Chapter  IV,  228 

of  Venus,  63,  174,  191,   194, 

234,  254 
relative,  n 
uniform,  26,  44,  108 
Mount  Wilson,  60,  264,  270 


NEWCOMB,  atmospheric  refrac- 
tion, 245 
figure  of  sun,  98 
mass  of  meteors,  106 
motion  of  Mercury,  62,  167 
motion  of  planets,   101,   172, 

236 

perturbations,  146 
NEWTON,  law  of  gravitation,  48, 

Chapter  III,  182 
laws  of  motion,  108 
theory  of  light,  7 
NICETAS,  75 


Orbit,  60,  113 
elements  of,  119 
of  Mercury,  61 
of  Venus,  63 


Index 


Orbital  velocities,  10,  34 
Oxford,  199,  201,  208 


PAINLEVE, 268 

Path,  21,  49 

Parabola,  21,  114,  115,  117 

Particles,  84 

Perihelion,  definition  of,  61,  161 

rotation  of,  61 
Period,  136 
PEROT,  shift  of  spectral  lines, 

271 
Perturbation,  128,  132,  141 

periodic,  144,  154 

second  order,  150 

secular,  144,  190 
PICARD,  measure  of  meridian,  83 
Plane,  of  ecliptic,  121 

of  orbit,  121 

Planetoids,  102,  166,  196,  236 
Precession,  143 

Principe,  66,  198,  201,  208,  233 
PTOLEMY,  73,  75 
PYTHAGORAS,  75 


RADCLIFFE,  97 

Reaction,  109 

Refraction,  abnormal,  207,  226 

atmospheric,  205,  242 

differential,  206 

formulas  of,  241 

index  of,  54,  266 

solar,  226,  240,  255 

through  ellipsoid,  247 
Relativity,  deductions,  49 

evidence    against,     Chapters 
VI,  VII,  VIII 

evidence  for,  Chapter  II 

general  theory,  44 

postulates  of,  19,  27,  23,  44, 
268 

restricted  principle  of,  26 

special  theory,  44 

theory  of,  Chapter  I 
Report,  67,  197 
Reversibility,  6 
Rigid  bodies,  31,  49 
Rotation,  26,  45 


RUSSELL,    star    displacements, 

220 


ST.  JOHN,  shift  of  spectral  lines, 

59,  272 

SCHUR,  figure  of  sun,  98 
SCHWARZSCHILD,  shift  of  spectral 

lines,  59 
DE  SITTER,  190 
Skin  friction,  262 
Sobral,  198,  202,  207,  209,  211, 

223,  239,  248 

Solar,  corona,  102,  201,  210,  241, 
256 

cycle,  99 

eclipse,  64,  Chapter  VII 

envelope,  23,  248 

gravity,  104 

spectrum,  57 

spots,  95 

temperature,  104 
Solar  System,  4 

stability  of,  147 
Space,  Chapter  I 

absolute,  20 

definition  of,  5 
Spectral  lines,  shift  of,  49,  56, 

270 

Spectroscope,  57 
Spectrum,  57 
Spheroids,  88 
Stars,  5 

shift  of,  212 

shooting,  1 06 

spectrum  of,  57 
Straight  line,  44,  108 
SWIFT,  discovery  of  Vulcan,  166 


Telescope,  79,  198 

horizontal,  200 
THOMSON,  138 
Tides,  138 
Time,  Chapter  I 

absolute,  6,  20 

definition  of,  5 

of  perihelion  passage,  121 
TISSERAND,    stability    of    solar 

system,  147 


Index 


277 


Trajectory,  22 

Transit,  of  Mercury,  155 

of  Venus,  97,  100,  202 

time  of,  168 

Trigonometric  series,  133 
Trigonometry,  discovery  of,  70 
TYCHO  BRAHE,  73,  76 


Variations,  132 
Vulcan,  62,  165 


WALMSLEY,  satellites  of  Jupiter, 

94,  230,  256 
Warp,  47,  68, 
Washington,  97 
WATSON,  discovery  of  Vulcan, 

1 66 


Zodiacal  light,    105,    181,   231, 
256 


Beginner's  Star=Book 

An  Easy  Guide  to  the  Stars  and  to  the  Astro- 
nomical Uses  of  the  Opera-Glass,  the 
Field-Glass,  and  the  Telescope 

By  Kelvin  McKready 


This  volume,  peculiarly  definite  and  helpful  in  method, 
is  especially  adapted  to  the  practical  needs  of  those  who 
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New  York         Q.   P.    Putnam's   Sons         London 


The  Outline  of  Science 

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